\begin{theorem}\label{global_existence} For $d=2, 3$,
there exists a diffeomorphism $\xi_0: \mathbb{R}^d \to \mathbb{R}^d$  {and a small constant $\varepsilon>0$} such that if the initial data $(\rho_0, u_0, F_0)$, with
$(\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\in H^3_x(\mathbb{R}^d)$, satisfies
{\begin{eqnarray*}\label{decay_estiwmates_1}
    \begin{cases}
      \rho_0(x) \mathrm{det}F_0(x)=1,\,\,\,\,F_0(x) = \nabla_y \xi_0(y),\,\,\,\,  y =\xi_0^{-1}(x), \ \forall x\in\mathbb{R}^d, \\[2mm]
      \|\rho_0(x) -1\|_{H^3_x} + \|u_0(x) -x\|_{H^3_x} + \|F_0(x) -I\|_{H^3_x} + \|\xi_0^{-1}(x)-x\|_{H^4_x}\le \varepsilon,
    \end{cases}
\end{eqnarray*}}then the Cauchy problem \eqref{elasticity_equation_1} admits a unique global strong solution
$(\rho, u, F)$ such that $\big(\rho(x,t) -\frac{1}{(1+t)^d}, u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big) \in C\big([0, \infty); H^3_x(\mathbb{R}^d)\big)$. Moreover, $(\rho, u, F)$ satisfies the following estimates:
\begin{eqnarray*}\label{decay_estimates_1}
  \begin{cases}
    \|\nabla_x^{i}(\rho(x,t) -\frac{1}{(1+t)^d})\|_{L^2_x}\le C\varepsilon(1+t)^{-\frac{d}{2}-\frac{1}{2}-i},\\[1mm]
     \|\nabla_x^{i}\big(u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big)\|_{L^2_x} \le C\varepsilon(1+t)^{\frac{d}{2}+\frac{1}{2}-i}
  \end{cases}
\end{eqnarray*}for all $t>0$ and $i=0,1,2,3$, {where $C$ is a generic positive constant depending on some known constants but independent of $\varepsilon$ and $t$.}
\end{theorem}

\label{section_1}
We consider the compressible {elastodynamics} in Eulerian coordinate \cite{Gurtin_1981,Dafermos_2010}:
\begin{eqnarray} \label{elasticity_equation_1}
    \begin{cases}
        \partial_t\rho+{\rm div} (\rho u) =0,\\
        \partial_t(\rho u)+{\rm div} (\rho u \otimes u)  + \nabla P(\rho)={\rm div}(\rho F F^\top),\\
        \partial_t F + u\cdot\nabla F=\nabla u F,\\
        (\rho,u,F)|_{t=0} = (\rho_0, u_0, F_0)
    \end{cases}
\end{eqnarray}
on $\mathbb{R}^d \times (0, +\infty)$, {with} $d = 2, 3${, where} $\rho=\rho(x,t)$, $u=u(x,t)$, $F=F(x,t)$ and $P(\rho)=R\rho^{\gamma}$ denotes the fluid density, the velocity field, the deformation gradient and the pressure, respectively. The $i$th component of ${\rm div}(\rho F F^\top)$ on the right-hand side of the momentum equation is $\partial_j(\rho F_{ik} F_{jk})$ and $(\nabla u F)_{ij}=\partial_k u_i F_{kj}$. The physical coefficients $R, \gamma$ satisfy $R>0$ and $\gamma>1$. Without loss of generality, we set $R=1$ throughout this paper. For smooth solutions $(\rho, u, F)$ of \eqref{elasticity_equation_1}, the physical constitution relation
\begin{align}\label{constitution_laws}
\begin{cases}
\rho\  \mathrm{det}F=1,\\
F_{j}\cdot\nabla F_{k}=F_{k}\cdot\nabla F_{j}, \quad 1\leq   j, k\leq d
\end{cases}
\end{align}
holds true for all $t\ge 0$, see \cite{Hu_2011, Qian_2010}. As a consequence of  \eqref{constitution_laws}, it is verified in \cite{Hu_2020, Hu_2020_1} that
\begin{align*}\label{constitution_laws_1}
{\rm div}(\rho F^\top)=0.
\end{align*}

We begin by briefly reviewing the literature on the well-posedness of elastodynamics {around a constant equilibrium}. In three-dimensional compressible elastodynamics, John \cite{John_1984} showed that singularities arise from arbitrarily small, spherically symmetric displacements in the absence of a null condition, while Tahvildar-Zadeh \cite{Tahvildar-Zadeh_1998} established a similar result for large displacements. In \cite{John_1988, Klainerman_1996}, the authors proved the existence of almost global solutions for three-dimensional quasilinear wave equations under the assumption of sufficiently small initial data. Significant progress in demonstrating the global existence of classical solutions was achieved independently by Sideris \cite{Sideris_1996,Sideris_2000} and Agemi \cite{Agemi_2000}, both assuming that the nonlinearity meets a null condition in three dimensions. For incompressible elastodynamics, particularly in 3D, the Hookean component of the system is inherently degenerate and satisfies a null condition, allowing Sideris and Thomases \cite{Sideris_2005,Sideris_2007} to establish global existence for this case. However, as noted by Lei \cite{Lei_2015}, the existence problem becomes more intricate in two dimensions, even with the null condition, since quadratic nonlinearities exhibit only critical time decay. The first significant result in tackling these issues was presented by Lei, Sideris, and Zhou \cite{Lei_2015}, who demonstrated the almost global existence of incompressible isotropic elastodynamics in 2D by reformulating the system in Eulerian coordinate. Lei \cite{Lei_2016} further leveraged the strong linear degeneracy of the nonlinearities in incompressible isotropic Hookean elastodynamics, which automatically satisfy a strong null condition, proving the global existence of classical solutions to the 2D Cauchy problem for small initial displacements in a specific weighted Sobolev space, with an alternative proof available in \cite{Wang_2017}. {All these references studied the stability issue of \eqref{elasticity_equation_1} around a non-zero constant equilibrium $$(\hat{\rho}, \hat{u}, \hat{F})=(1,0, I).$$ Indeed, the linearized system of \eqref{elasticity_equation_1}
at $(1, 0, I)$ is a coupling wave system for shear waves and pressure waves with two different wave speeds, where $I$ stands for the identity matrix of order $d$. For incompressible models, the pressure wave disappears, and the linearized system for incompressible elastodynamics takes the form
\begin{equation}\label{wave}
\partial_t^2 \omega-\Delta \omega=0,
\end{equation}
 where $\omega$ is either the vorticity of velocity $u$ or the curl of $F$. The stability around a constant equilibrium is justified by following the well posedness theories for quasilinear wave equations via either the vector field method \cite{Sideris_2005, Lei_2016} or the space-time resonance method \cite{Wang_2017}.}

{Different from the stability around a constant equilibrium, there is an extensive range of interests on the stability of \eqref{elasticity_equation_1} around some non-constant equilibrium in
the communities of both physics and mathematics. As an example of non-constant equilibrium, in }\cite{Sideris_2017}, an affine motion is defined as a one-parameter family of deformations of the form
$$x(y,t)=A(t)y,$$ where $x(y,t)$ is the flow map with the initial position $y\in\mathbb{R}^d$. For affine motions, all hydrodynamical quantities are expressed explicitly in terms of the deformation gradient $A(t)$ and the initial conditions. The authors in \cite{Hu_preprint} {studied the stability of} an affine solution for the 3D compressible viscoelasticity system in Eulerian coordinate, given by
\begin{equation}\label{1-1}
    \bar{\rho}=\frac{1}{(1+t)^{3}}, \bar{u} = \frac{x}{1+t},  \bar{F}= (1+t)I,
  \end{equation}
 with the initial data $(1, x, I).$ {The magnitude of the deformation gradient in \eqref{1-1} grows at a rate proportional to time, and that deformation gradient does not lead to finite time collapse or blow-up of the flow. The stability of the affine solution \eqref{1-1} is justified in \cite{Hu_preprint} for compressible viscoelasticity under small $H^2$-perturbations. The key difficulty lies in the fact the coefficients of the linearised system in \cite{Hu_preprint} depends on the spatial and temporal variables. A natural question then arises: does such stability persist for compressible elastodynamics \eqref{elasticity_equation_1} when the viscosity is absent. In this work, we provide an affirmative answer
 by establishing}

We begin by briefly reviewing the literature on the well-posedness of elastodynamics {around a constant equilibrium}. In three-dimensional compressible elastodynamics, John \cite{John_1984} showed that singularities arise from arbitrarily small, spherically symmetric displacements in the absence of a null condition, while Tahvildar-Zadeh \cite{Tahvildar-Zadeh_1998} established a similar result for large displacements. In \cite{John_1988, Klainerman_1996}, the authors proved the existence of almost global solutions for three-dimensional quasilinear wave equations under the assumption of sufficiently small initial data. Significant progress in demonstrating the global existence of classical solutions was achieved independently by Sideris \cite{Sideris_1996,Sideris_2000} and Agemi \cite{Agemi_2000}, both assuming that the nonlinearity meets a null condition in three dimensions. For incompressible elastodynamics, particularly in 3D, the Hookean component of the system is inherently degenerate and satisfies a null condition, allowing Sideris and Thomases \cite{Sideris_2005,Sideris_2007} to establish global existence for this case. However, as noted by Lei \cite{Lei_2015}, the existence problem becomes more intricate in two dimensions, even with the null condition, since quadratic nonlinearities exhibit only critical time decay. The first significant result in tackling these issues was presented by Lei, Sideris, and Zhou \cite{Lei_2015}, who demonstrated the almost global existence of incompressible isotropic elastodynamics in 2D by reformulating the system in Eulerian coordinate. Lei \cite{Lei_2016} further leveraged the strong linear degeneracy of the nonlinearities in incompressible isotropic Hookean elastodynamics, which automatically satisfy a strong null condition, proving the global existence of classical solutions to the 2D Cauchy problem for small initial displacements in a specific weighted Sobolev space, with an alternative proof available in \cite{Wang_2017}. {All these references studied the stability issue of \eqref{elasticity_equation_1} around a non-zero constant equilibrium $$(\hat{\rho}, \hat{u}, \hat{F})=(1,0, I).$$ Indeed, the linearized system of \eqref{elasticity_equation_1}
at $(1, 0, I)$ is a coupling wave system for shear waves and pressure waves with two different wave speeds, where $I$ stands for the identity matrix of order $d$. For incompressible models, the pressure wave disappears, and the linearized system for incompressible elastodynamics takes the form
\begin{equation}\label{wave}
\partial_t^2 \omega-\Delta \omega=0,
\end{equation}
 where $\omega$ is either the vorticity of velocity $u$ or the curl of $F$. The stability around a constant equilibrium is justified by following the well posedness theories for quasilinear wave equations via either the vector field method \cite{Sideris_2005, Lei_2016} or the space-time resonance method \cite{Wang_2017}.}

{Different from the stability around a constant equilibrium, there is an extensive range of interests on the stability of \eqref{elasticity_equation_1} around some non-constant equilibrium in
the communities of both physics and mathematics. As an example of non-constant equilibrium, in }\cite{Sideris_2017}, an affine motion is defined as a one-parameter family of deformations of the form
$$x(y,t)=A(t)y,$$ where $x(y,t)$ is the flow map with the initial position $y\in\mathbb{R}^d$. For affine motions, all hydrodynamical quantities are expressed explicitly in terms of the deformation gradient $A(t)$ and the initial conditions. The authors in \cite{Hu_preprint} {studied the stability of} an affine solution for the 3D compressible viscoelasticity system in Eulerian coordinate, given by
\begin{equation}\label{1-1}
 \bar{\rho}=\frac{1}{(1+t)^{3}}, \bar{u} = \frac{x}{1+t}, \bar{F}= (1+t)I,
 \end{equation}
 with the initial data $(1, x, I).$ {The magnitude of the deformation gradient in \eqref{1-1} grows at a rate proportional to time, and that deformation gradient does not lead to finite time collapse or blow-up of the flow. The stability of the affine solution \eqref{1-1} is justified in \cite{Hu_preprint} for compressible viscoelasticity under small $H^2$-perturbations. The key difficulty lies in the fact the coefficients of the linearised system in \cite{Hu_preprint} depends on the spatial and temporal variables. A natural question then arises: does such stability persist for compressible elastodynamics \eqref{elasticity_equation_1} when the viscosity is absent. In this work, we provide an affirmative answer
 by establishing}

{With the absence of the elastic stress $\textrm{div}(\rho FF^\top)$ in the momentum equation, the system \eqref{elasticity_equation_1} becomes a degenerate system whenever the physical vacuum is present. Indeed, by globally solving a Hamiltonian system of ordinary differential equations for the deformation gradient, the free boundary problem of the three dimensional compressible and incompressible Euler equations with an affine initial condition and physical vacuum boundary condition is constructed in \cite{Sideris_2017}. Hadzic and Jang in \cite{Hadzic_2018} has verified the global-in-time nonlinear stability in Sobolev spaces of the affine solutions to compressible Euler equations for $\gamma\in(1,\frac53]$, and the full range of $\gamma\in (1,\infty)$ is studied in \cite{Shkoller_2019}. Note that the determinant of deformation gradients of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} grows at the maximal rate of $(1+t)^3$, as $t\rightarrow\infty$. In this sense, Theorem \ref{global_existence} would be regarded as the asymptotic nonlinear stability of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} when the elastic stress is taken into consideration in the momentum equation.

\section{Global well-posedness in Eulerian coordinate}\label{section_4}
This section is devoted to the completion of the proof of Theorem \ref{global_existence}. Throughout this section, C denotes a generic positive constant that depends on
some known constants, but independent of $\varepsilon$, $\varepsilon_1$ and $t$.
\subsection{Regularity estimates}\label{subsection_7}
With Theorem \ref{global_existence_Lagrangian}, we are going to construct a global-in-time solution to the system
\eqref{elasticity_equation_1} in Eulerian coordinate. For any $(u_0, \xi_0)$ given in Theorem \ref{global_existence}, we define $V_0$ as follows
 \begin{equation}\label{V_0}
 V_0(y)=u_0(\xi_0(y)).
\end{equation}
{Combining \eqref{V_0} with the assumptions of Theorem \ref{global_existence}, we deduce
\begin{equation*}\label{1}
 \begin{split}
 \|V_0(y)-y\|_{H^3_y} + \|\xi_0(y) -y\|_{H^4_y}\le C\varepsilon + C\varepsilon^3 \le C\varepsilon.
 \end{split}
\end{equation*}
Then, letting $\varepsilon = \frac{\varepsilon_1}{C}$ yields
\begin{equation}\label{2}
 \begin{split}
 \|V_0(y)-y\|_{H^3_y} + \|\xi_0(y) -y\|_{H^4_y}\le \varepsilon_1.
 \end{split}
\end{equation}
Substituting $V_0(y)$ and $\xi_0(y)$, that satisfy \eqref{V_0} and \eqref{2}, into Theorem \ref{global_existence_Lagrangian}, we obtain the existence of a unique global solution $(\xi,V)$ to the system \eqref{compressible_elasticity_equation_3}.

For the second term, we similarly obtain
 \begin{equation}\label{Euler_part_17}
 \begin{split}
 &\int_{\mathbb{R}^d} \Big|{\rm adj}[\nabla_y \xi] - (1+t)^{d-1}I\Big|^2\,dy\\ &\le C \Big(\|\nabla_y \xi\|_{L^\infty_y}^{2(d-2)} + (1+t)^{2(d-2)}\Big)\|\nabla_y \xi -(1+t)I\|_{L^2_y}^2\\ & \le C\varepsilon^2 (1+t)^{2d-3}.
 \end{split}
 \end{equation}
Substituting \eqref{Euler_part_16} and \eqref{Euler_part_17} into \eqref{Euler_part_14} and \eqref{Euler_part_15} gives
 \begin{equation}\label{Euler_part_18}
 \begin{split}
 \|\rho(x,t) -\frac{1}{(1+t)^d}\|_{L^2_x}^2 \le C\varepsilon^2 (1+t)^{-d-1}.
 \end{split}
 \end{equation}
 Next, we consider the estimate of $\|\nabla \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}$ and $\|\nabla^2 \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}$. Combining \eqref{decay_estimates_Lagrangian}, \eqref{Euler_part_4_1}, \eqref{Euler_part_nn_2}-\eqref{Euler_part_nn_3}, {$\varepsilon = \frac{\varepsilon_1}{C}$} with $\nabla_y B_{ij}= -B_{il}\partial_l\nabla_y\xi_{k}B_{kj}$ and $\nabla_y \mathcal{J}= \mathcal{J} B_{ij}\partial_j\nabla_y \xi_{i}$ in \eqref{flow_map_5_1}, we have
 \begin{equation}\label{Euler_part_10}
 \begin{split}
 &\|\nabla_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}^2 \\ &\le C(1+t)^d \int_{\mathbb{R}^d} |B\nabla_y(\mathcal{J}^{-1})|^2\,dy\\
 &\le C(1+t)^d\int_{\mathbb{R}^d} |B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2\,dy\\
 &\le C(1+t)^d|B|^4 |\mathcal{J}|^{-2} \int_{\mathbb{R}^d} |\nabla^2_y \xi|^2\,dy\\
 &\le C \varepsilon^2 (1+t)^{-d-3}
 \end{split}
 \end{equation}
 and
 \begin{equation}\label{Euler_part_11}
 \begin{split}
 &\|\nabla^2_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}^2\\
 &\le C(1+t)^d\int_{\mathbb{R}^d} \big|B\nabla_y[B\nabla_y(\mathcal{J}^{-1})]\big|^2\,dy \\
 &\le C(1+t)^d\Big[\int_{\mathbb{R}^d} |B\nabla_y B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2 \,dy + \int_{\mathbb{R}^d} |BB \mathcal{J}^{-3}\nabla_y\mathcal{J}\nabla_y\mathcal{J}|^2 \,dy\\
 & +\int_{\mathbb{R}^d} |BB\mathcal{J}^{-2}\nabla_y^2\mathcal{J}|^2 \,dy\Big]\\
 &\le C \varepsilon^2 (1+t)^{-d-6} + C \varepsilon^2 (1+t)^{-d-6} +C \varepsilon^2 (1+t)^{-d-5} \\
 &\le C \varepsilon^2 (1+t)^{-d-5}.
 \end{split}
 \end{equation}
 Finally, we estimate $\|\nabla^3_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}$. Combining \eqref{decay_estimates_Lagrangian}, \eqref{Euler_part_4_1}, \eqref{Euler_part_nn_4}, $\varepsilon = \frac{\varepsilon_1}{C}$ with $\nabla_y B_{ij}= -B_{il}\partial_l\nabla_y\xi_{k}B_{kj}$ and $\nabla_y \mathcal{J}= \mathcal{J} B_{ij}\partial_j\nabla_y \xi_{i}$ in \eqref{flow_map_5_1}, we have
 \begin{align}\label{Euler_part_11_1}
 &\notag\|\nabla^3_x \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}^2\\ \notag &\le C(1+t)^d\int_{\mathbb{R}^d} \Big|B\nabla_y\big\{B\nabla_y\big[B\nabla_y(\mathcal{J}^{-1})\big]\big\}\Big|^2\,dy\\ &
 \le C(1+t)^d\Big[\int_{\mathbb{R}^d} |B\nabla_y B\nabla_y B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2 \,dy + \int_{\mathbb{R}^d} |BB \nabla_y^2 B\mathcal{J}^{-2}\nabla_y\mathcal{J}|^2 \,dy\notag\\ & +\int_{\mathbb{R}^d} |BB\nabla_y B\mathcal{J}^{-3}\nabla_y\mathcal{J}\nabla_y\mathcal{J}|^2 \,dy +\int_{\mathbb{R}^d} |BB\nabla_y B\mathcal{J}^{-2}\nabla_y^2\mathcal{J}|^2 \,dy\\ & +\int_{\mathbb{R}^d} |BBB\mathcal{J}^{-4}\nabla_y\mathcal{J}\nabla_y\mathcal{J}\nabla_y\mathcal{J}|^2 \,dy +\int_{\mathbb{R}^d} |BBB\mathcal{J}^{-3}\nabla_y\mathcal{J}\nabla_y^2\mathcal{J}|^2 \,dy\notag\\ & \notag +\int_{\mathbb{R}^d} |BBB\mathcal{J}^{-2}\nabla_y^3\mathcal{J}|^2\,dy\Big]\\ &\notag
 \le\, C \varepsilon^2 (1+t)^{-d-9} + C \varepsilon^2 (1+t)^{-d-8} + C \varepsilon^2 (1+t)^{-d-9}+ C \varepsilon^2 (1+t)^{-d-8} \notag \\ &+ C \varepsilon^2 (1+t)^{-d-9} + C \varepsilon^2 (1+t)^{-d-8} + C \varepsilon^2 (1+t)^{-d-7}\notag\\ &\le \,C \varepsilon^2 (1+t)^{-d-7}.\notag
\end{align}
 From \eqref{Euler_part_ww_1}-\eqref{Euler_part_ww_4}, \eqref{Euler_part_ee_1}-\eqref{Euler_part_ee_4}, and \eqref{Euler_part_18}-\eqref{Euler_part_11_1}, we conclude that
 \begin{eqnarray*}\label{decay_1}
 \begin{cases}
 \|\nabla_x^i \big(\rho(x,t) -\frac{1}{(1+t)^d}\big)\|_{L^2_x}\le C\varepsilon (1+t)^{-\frac{d}{2}-\frac{1}{2}-i},\\[1mm]
 \|\nabla_x^i\big(u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big)\|_{L^2_x} \le C\varepsilon(1+t)^{\frac{d}{2}+\frac{1}{2}-i}
 \end{cases}
 \end{eqnarray*}
 for all $t \ge 0$ and $i=0,1,2,3$. The proof of Theorem \ref{global_existence} is complete.

We prefer to work in the framework of $H^3$-spaces in order to keep the strategy as clear as possible. With our strategy in hand, it is possible to reduce the regularity of solutions to some Besov spaces, however the presentation would be more technical and complicated, and we leave it for the interested readers.

{With the absence of the elastic stress $\textrm{div}(\rho FF^\top)$ in the momentum equation, the system \eqref{elasticity_equation_1} becomes a degenerate system whenever the physical vacuum is present. Indeed, by globally solving a Hamiltonian system of ordinary differential equations for the deformation gradient, the free boundary problem of the three dimensional compressible and incompressible Euler equations with an affine initial condition and physical vacuum boundary condition is constructed in \cite{Sideris_2017}. Hadzic and Jang in \cite{Hadzic_2018} has verified the global-in-time nonlinear stability in Sobolev spaces of the affine solutions to compressible Euler equations for $\gamma\in(1,\frac53]$, and the full range of $\gamma\in (1,\infty)$ is studied in \cite{Shkoller_2019}. Note that the determinant of deformation gradients of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} grows at the maximal rate of $(1+t)^3$, as $t\rightarrow\infty$. In this sense, Theorem \ref{global_existence} would be regarded as the asymptotic nonlinear stability of the affine solution in \cite{Sideris_2017, Hadzic_2018, Shkoller_2019} when the elastic stress is taken into consideration in the momentum equation.

The presence of the elastic stress $\textrm{div}(\rho FF^\top)$ in the momentum equation makes the wave operator non-degenerate. Motivated by the study of compressible Euler equations in \cite{Shkoller_2019}, we turn to employing a Lagrangian transformation and consider a different class of affine solutions that nevertheless share the same scaling property, namely $\det A(t)\sim (1+t)^3$. Compared with the Euler equations, the presence of elastic stress introduces an additional diffusion mechanism, see~\eqref{compressible_elasticity_equation_5}. However, this feature requires a different analytical framework, and the framework developed in \cite{Shkoller_2019} cannot be applied in a straightforward manner. Fortunately, the expanding factor on $t$ in the affine solution \eqref{1-1} contributes a damping term for the velocity in linearized system of \eqref{elasticity_equation_1} in Lagrangian coordinate, see \eqref{compressible_elasticity_equation_5}. However, due to the expanding factor in $t$ of the affine solution,  the coefficients of the linearized system of  \eqref{elasticity_equation_1} involves functions depending on $t$. This kind of
hyperbolic systems with variable coefficients excludes the applicability of both the vector field method and the space-time resonance method, which are designed for quasilinear wave equations with constant variables. In order to overcome this difficulty, we turn to the $t-$dependent damping effect of the velocity. Indeed, with this damping at hand, different weights for unknowns will be constructed in the form of $\mathcal{L}(t)$, and  the differential inequality \eqref{est_L_1} is verified. More precisely, since we are concerned with the stability of a non-constant equilibrium, it is necessary to introduce different temporal weights in a series of uniform estimates. For instance, the terms in $\mathcal{L}(t)$ in \eqref{L} carry distinct weights {depending on $\gamma$ and the dimension}. Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all {unknowns} become uniform. This uniformity plays a key role in establishing the global well-posedness of the compressible elastodynamics system in both two and three dimensions for all $\gamma>1$ within a unified framework. As a consequence, the global existence in Lagrangian coordinate would be established in Theorem \ref{global_existence_Lagrangian} with the help of }the local well-posedness  and the global-in-time estimate of the quantity $\mathcal{L}(t)$.

The global well-posedness in Lagrangian coordinate will be transferred back into the Eulerian coordinate as stated in Theorem \ref{global_existence} via the flow map. We remark that in Lagrangian coordinate, the impact of differentiation on time decay is largely concealed, whereas in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$. Moreover, the $L^2$ norms of $\rho(x,t) -\frac{1}{(1+t)^d}$, $u(x,t)-\frac{x}{1+t}$ and  $F(x,t) -(1+t)I$ decay at faster rates compared with those obtained in \cite{Hu_preprint}.

The rest of the paper is organized as follows. In Section \ref{section_3},
we will reformulate the Cauchy problem \eqref{elasticity_equation_1} in the Lagrangian coordinate and establish the global well-posedness of system \eqref{compressible_elasticity_equation_5} through energy estimates. Building upon Theorem \ref{global_existence_Lagrangian}, Section \ref{section_4} is devoted to the construction of the global solution of system \eqref{elasticity_equation_1}
and the proof of Theorem \ref{global_existence}.

To prove Theorem~\ref{global_existence}, the authors:
\begin{itemize}
\item \emph{Employ a Lagrangian transformation} ("we turn to employing a Lagrangian transformation") and "consider a different class of affine solutions that nevertheless share the same scaling property, namely $\det A(t)\sim (1+t)^3$."
\item Use that the affine expansion yields a \emph{$t$-dependent damping} in the Lagrangian linearized system ("the expanding factor on $t$ in the affine solution \eqref{1-1} contributes a damping term for the velocity in linearized system ... see \eqref{compressible_elasticity_equation_5}").
\item Since the coefficients depend on $t$ ("the coefficients ... involves functions depending on $t$"), they do not use vector-field/space-time resonance methods; instead they "turn to the $t$-dependent damping effect of the velocity" and construct "different weights for unknowns ... in the form of $\mathcal{L}(t)$" and verify "the differential inequality \eqref{est_L_1}."
\item Introduce "different temporal weights in a series of uniform estimates"; "the terms in $\mathcal{L}(t)$ in \eqref{L} carry distinct weights". "Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all unknowns become uniform," and this "plays a key role in establishing the global well-posedness ... in both two and three dimensions". This yields global existence in Lagrangian variables ("the global existence in Lagrangian coordinate would be established in Theorem~\ref{global_existence_Lagrangian} with the help of the local well-posedness and the global-in-time estimate of the quantity $\mathcal{L}(t)$").
\item \emph{Transfer back to Eulerian coordinates} ("transferred back into the Eulerian coordinate as stated in Theorem~\ref{global_existence} via the flow map"), and obtain the stated decay, noting that "in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$."
\end{itemize}

To prove the main theorem, the authors:
\begin{itemize}
\item \emph{Employ a Lagrangian transformation} ("we turn to employing a Lagrangian transformation") and "consider a different class of affine solutions that nevertheless share the same scaling property, namely $\det A(t)\sim (1+t)^3$."
\item Use that the affine expansion yields a \emph{$t$-dependent damping} in the Lagrangian linearized system ("the expanding factor on $t$ in the affine solution
\begin{equation}\label{1-1}
    \bar{\rho}=\frac{1}{(1+t)^{3}}, \bar{u} = \frac{x}{1+t},  \bar{F}= (1+t)I,
  \end{equation}
contributes a damping term for the velocity in linearized system ... see
\begin{eqnarray} \label{compressible_elasticity_equation_5}
	\begin{cases}
		\partial_t \widetilde{\eta}=v,\\
		\partial_t v_{i} + \frac{2}{1+t} v_{i} + \frac{1}{(1+t)^{d(\gamma -1)+2}} a_{il}\partial_l q -\Delta \widetilde{\eta}_{i} =0,\\
		(v,\widetilde{\eta})|_{t=0} = \big(v_0, \widetilde{\eta}_0\big)
	\end{cases}
  \end{eqnarray}
).
\item Since the coefficients depend on $t$ ("the coefficients ... involves functions depending on $t$"), they do not use vector-field/space-time resonance methods; instead they "turn to the $t$-dependent damping effect of the velocity" and construct "different weights for unknowns ... in the form of $\mathcal{L}(t)$" and verify the differential inequality
\begin{equation}\label{est_L_1}
  \begin{split}
  \udt \mathcal{L}(t) + \frac{1}{1+t} \mathcal{L}(t)\le \frac{C}{(1+t)^{2}}\big(\mathcal{L}(t) + \mathcal{L}^{\frac{7}{2}}(t)\big).
  \end{split}
\end{equation}
\item Introduce "different temporal weights in a series of uniform estimates"; "the terms in $\mathcal{L}(t)$ in
\begin{equation}\label{L}
	\begin{split}
    \mathcal{L}(t) := &\|v(t)\|_{H^3}^2 + \frac{2}{(1+t)^2}\|\widetilde{\eta}(t)\|_{H^3}^2 + \|\nabla \widetilde{\eta}(t)\|_{H^3}^2+ \frac{2}{1+t}\sum_{i = 0}^{3} \langle\nabla^{i} v(t),\nabla^{i} \widetilde{\eta}(t)\rangle \\ &+ \frac{1}{\gamma(1+t)^{d(\gamma -1)+2}} \int_{\mathbb{R}^d}J^{\gamma+1}(t) |\nabla^3 q(t)|^2 \,dy.
	\end{split}
\end{equation}
carry distinct weights". "Thanks to the presence of the deformation gradient, which provides an effective diffusion mechanism, the temporal weights of all derivatives of all unknowns become uniform," and this "plays a key role in establishing the global well-posedness ... in both two and three dimensions". This yields global existence in Lagrangian variables; more precisely, they establish the following theorem.

\begin{theorem}\label{global_existence_Lagrangian}
       For $d=2,3$,  there exists a sufficiently small  constant $\varepsilon_1>0$ such that the Cauchy problem \eqref{compressible_elasticity_equation_3} admits a global strong solution
       $(V,\xi)$ satisfying $(V(y,t)-y, \xi(y,t) - (1+t)y)$ in $C\big([0, \infty); H^3_y(\mathbb{R}^d)\times H^4_y(\mathbb{R}^d)\big)$,  provided that
          \begin{equation*} \label{initial_assumption_Lagrangian}
              \begin{split}
              \|V_0(y)-y\|_{H^3_y} + \|\xi_0(y) -y\|_{H^4_y}\le \varepsilon_1.
              \end{split}
          \end{equation*}
Moreover, $(V,\xi)$ satisfies the following estimates:
\begin{eqnarray}\label{decay_estimates_Lagrangian}
  \begin{cases}
    \|V(y,t) -y\|_{H^3_y} \le C\varepsilon_1(1+t)^{\frac{1}{2}},\\[1mm]
    \|\xi(y,t) - (1+t)y\|_{H^3_y} \le C\varepsilon_1(1+t)^{\frac{3}{2}},\\[1mm]
     \|\nabla_y \xi(y,t) - (1+t)I\|_{H^3_y} \le C\varepsilon_1(1+t)^{\frac{1}{2}}
  \end{cases}
\end{eqnarray}
for all $t>0$, where $C$ is a generic positive constant depending on some known constants but independent of $\varepsilon_1$ and $t$.
\end{theorem}

\item \emph{Transfer back to Eulerian coordinates} ("transferred back into the Eulerian coordinate as stated in the main theorem via the flow map"), and obtain the stated decay, noting that "in Eulerian coordinate each additional derivative accelerates decay by a factor of $(1+t)^{-1}$."
\end{itemize}

\label{global_existence} For $d=2, 3$,
there exists a diffeomorphism $\xi_0: \mathbb{R}^d \to \mathbb{R}^d$  {and a small constant $\varepsilon>0$} such that if the initial data $(\rho_0, u_0, F_0)$, with
$(\rho_0(x) -1, u_0(x) -x, F_0(x) -I)\in H^3_x(\mathbb{R}^d)$, satisfies
{\begin{eqnarray*}\label{decay_estiwmates_1}
    \begin{cases}
      \rho_0(x) \mathrm{det}F_0(x)=1,\,\,\,\,F_0(x) = \nabla_y \xi_0(y),\,\,\,\,  y =\xi_0^{-1}(x), \ \forall x\in\mathbb{R}^d, \\[2mm]
      \|\rho_0(x) -1\|_{H^3_x} + \|u_0(x) -x\|_{H^3_x} + \|F_0(x) -I\|_{H^3_x} + \|\xi_0^{-1}(x)-x\|_{H^4_x}\le \varepsilon,
    \end{cases}
\end{eqnarray*}}then, in establishing the main theorem, the Cauchy problem
\begin{eqnarray} \label{elasticity_equation_1}
    \begin{cases}
        \partial_t\rho+{\rm div} (\rho u) =0,\\
        \partial_t(\rho u)+{\rm div} (\rho u \otimes u)  + \nabla P(\rho)={\rm div}(\rho F F^\top),\\
        \partial_t F + u\cdot\nabla F=\nabla u F,\\
        (\rho,u,F)|_{t=0} = (\rho_0, u_0, F_0)
    \end{cases}
\end{eqnarray}
admits a unique global strong solution
$(\rho, u, F)$ such that $\big(\rho(x,t) -\frac{1}{(1+t)^d}, u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big) \in C\big([0, \infty); H^3_x(\mathbb{R}^d)\big)$. Moreover, $(\rho, u, F)$ satisfies the following estimates:
\begin{eqnarray*}\label{decay_estimates_1}
  \begin{cases}
    \|\nabla_x^{i}(\rho(x,t) -\frac{1}{(1+t)^d})\|_{L^2_x}\le C\varepsilon(1+t)^{-\frac{d}{2}-\frac{1}{2}-i},\\[1mm]
     \|\nabla_x^{i}\big(u(x,t)-\frac{x}{1+t}, F(x,t) -(1+t)I\big)\|_{L^2_x} \le C\varepsilon(1+t)^{\frac{d}{2}+\frac{1}{2}-i}
  \end{cases}
\end{eqnarray*}for all $t>0$ and $i=0,1,2,3$, {where $C$ is a generic positive constant depending on some known constants but independent of $\varepsilon$ and $t$.}

Question: For each dimension d ∈ {2,3}, consider the compressible elastodynamics Cauchy problem on R^d given by ∂tρ + div(ρu) = 0, ∂t(ρu) + div(ρu ⊗ u) + ∇P(ρ) = div(ρFF^T), ∂tF + u·∇F = (∇u)F, with initial data (ρ,u,F)|_{t=0} = (ρ0,u0,F0). Assume (ρ0(x)−1, u0(x)−x, F0(x)−I) ∈ H^3_x(R^d), where I is the d×d identity matrix, and if ξ0:R^d→R^d is a diffeomorphism then y = ξ0^{-1}(x). Which existence and decay statement is guaranteed for small perturbations of the expanding affine state ((1+t)^{-d}, x/(1+t), (1+t)I)?

For each d ∈ {2,3}, there exist a diffeomorphism ξ0:R^d→R^d and a sufficiently small constant ε > 0 such that, if the initial data satisfy ρ0(x) det F0(x) = 1 and F0(x) = ∇y ξ0(y) with y = ξ0^{-1}(x) for all x ∈ R^d, together with the smallness condition ||ρ0−1||_{H^3_x} + ||u0−x||_{H^3_x} + ||F0−I||_{H^3_x} + ||ξ0^{-1}−x||_{H^4_x} ≤ ε, then the Cauchy problem admits a unique global strong solution (ρ,u,F) such that (ρ(x,t)−(1+t)^{-d}, u(x,t)−x/(1+t), F(x,t)−(1+t)I) belongs to C([0,∞); H^3_x(R^d)). Moreover, for every t > 0 and i = 0,1,2,3, the estimates ||∇_x^i(ρ(x,t)−(1+t)^{-d})||_{L^2_x} ≤ Cε(1+t)^(−d/2−1/2−i) and ||∇_x^i(u(x,t)−x/(1+t), F(x,t)−(1+t)I)||_{L^2_x} ≤ Cε(1+t)^(d/2+1/2−i) hold, where C is a positive constant depending only on fixed known constants and independent of ε and t.

Choices:
B. For each d ∈ {2,3}, there exist a diffeomorphism ξ0:R^d→R^d and a sufficiently small constant ε > 0 such that, if the initial data satisfy ρ0(x) det F0(x) = 1 and F0(x) = ∇y ξ0(y) with y = ξ0^{-1}(x) for all x ∈ R^d, together with the smallness condition ||ρ0−1||_{H^3_x} + ||u0−x||_{H^3_x} + ||F0−I||_{H^3_x} + ||ξ0^{-1}−x||_{H^4_x} ≤ ε, then the Cauchy problem admits a unique global strong solution (ρ,u,F) such that (ρ(x,t)−(1+t)^{-d}, u(x,t)−x/(1+t), F(x,t)−(1+t)I) belongs to C([0,∞); H^3_x(R^d)). Moreover, for every t > 0 and i = 0,1,2,3, the estimates ||∇_x^i(ρ(x,t)−(1+t)^{-d})||_{L^2_x} ≤ Cε(1+t)^(−d/2−i) and ||∇_x^i(u(x,t)−x/(1+t), F(x,t)−(1+t)I)||_{L^2_x} ≤ Cε(1+t)^(d/2−i) hold, where C is a positive constant depending only on fixed known constants and independent of ε and t.
C. For each d ∈ {2,3}, there exist a diffeomorphism ξ0:R^d→R^d and a sufficiently small constant ε > 0 such that, if the initial data satisfy ρ0(x) det F0(x) = 1 and F0(x) = ∇y ξ0(y) with y = ξ0^{-1}(x) for all x ∈ R^d, together with the smallness condition ||ρ0−1||_{H^3_x} + ||u0−x||_{H^3_x} + ||F0−I||_{H^3_x} + ||ξ0^{-1}−x||_{H^4_x} ≤ ε, then the Cauchy problem admits a unique global strong solution (ρ,u,F) such that (ρ(x,t)−(1+t)^{-d}, u(x,t)−x/(1+t), F(x,t)−(1+t)I) belongs to C([0,∞); H^3_x(R^d)).
D. For each d ∈ {2,3}, there exists a sufficiently small constant ε > 0 such that for every diffeomorphism ξ0:R^d→R^d, if the initial data satisfy ρ0(x) det F0(x) = 1 and F0(x) = ∇y ξ0(y) with y = ξ0^{-1}(x) for all x ∈ R^d, together with the smallness condition ||ρ0−1||_{H^3_x} + ||u0−x||_{H^3_x} + ||F0−I||_{H^3_x} + ||ξ0^{-1}−x||_{H^4_x} ≤ ε, then the Cauchy problem admits a unique global strong solution (ρ,u,F) such that (ρ(x,t)−(1+t)^{-d}, u(x,t)−x/(1+t), F(x,t)−(1+t)I) belongs to C([0,∞); H^3_x(R^d)). Moreover, for every t > 0 and i = 0,1,2,3, the estimates ||∇_x^i(ρ(x,t)−(1+t)^{-d})||_{L^2_x} ≤ Cε(1+t)^(−d/2−1/2−i) and ||∇_x^i(u(x,t)−x/(1+t), F(x,t)−(1+t)I)||_{L^2_x} ≤ Cε(1+t)^(d/2+1/2−i) hold, where C is a positive constant depending only on fixed known constants and independent of ε, t, and ξ0.
E. For each d ∈ {2,3}, there exist a diffeomorphism ξ0:R^d→R^d and a sufficiently small constant ε > 0 such that, if the initial data satisfy ρ0(x) det F0(x) = 1 and F0(x) = ∇y ξ0(y) with y = ξ0^{-1}(x) for all x ∈ R^d, together with the smallness condition ||ρ0−1||_{H^3_x} + ||u0−x||_{H^3_x} + ||F0−I||_{H^3_x} + ||ξ0^{-1}−x||_{H^4_x} ≤ ε, then the Cauchy problem admits a unique global strong solution (ρ,u,F) such that (ρ(x,t)−(1+t)^{-d}, u(x,t)−x/(1+t), F(x,t)−(1+t)I) belongs to C([0,∞); H^3_x(R^d)). Moreover, for every t > 0 and i = 0,1,2,3, the estimates ||∇_x^i(ρ(x,t)−(1+t)^{-d})||_{L^2_x} ≤ Cε(1+t)^(−d/2−1/2−i) and ||∇_x^i(u(x,t)−x/(1+t), F(x,t)−(1+t)I)||_{L^2_x} ≤ Cε(1+t)^(−d/2−1/2−i) hold, where C is a positive constant depending only on fixed known constants and independent of ε and t.

Eulerian derivative gains exactly one extra (1+t)^{-1} factor while retaining the additional 1/2-decay offset

dropped the quantitative decay estimates while keeping the global existence and H^3-continuity conclusion

quantifier order and dependence on the admissible initial diffeomorphism ξ0

distinct decay behavior of density versus velocity/deformation inherited from the weighted Lagrangian energy and Eulerian transfer


\begin{example}\label{example}
	Let $M^n$ be a closed aspherical spin manifold with $n\ge 5$, e.g., the $n$-torus. Suppose that $\pi_1(M)$ is represented by finitely many disjoint embedded circles $S^1$ in $M$, whose tubular neighborhoods are diffeomorphic to $S^1 \times B^{n-1}$. Remove these tubular neighborhoods and attach $B^2 \times S^{n-2}$ to obtain a closed manifold $M_1$. Then $M_1$ is simply connected.

	Finally, set $Z \coloneqq M_1 \# M_1$, which is a spin surgery of $M$. Note that $Z$ is simply connected, and the Dirac operator on $Z$ has vanishing $KO$-index as $Z$ is a double. Hence, $Z$ admits a metric with positive scalar curvature \cite{Stolz}.
\end{example}

The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \cite{ChodoshLi,Gromov5fold}.

The non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \cite{WangZhang} and Su \cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \cite{ChodoshLi,ChenChuZhu,MR4291609}.

In this paper, we extend the above non-existence of positive scalar metrics to a significantly broader class of manifolds.

\begin{abstract}
 Let $M$ be a closed aspherical manifold. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$. We show that on any spin surgery of $M$ along a region whose induced homomorphism on the fundamental group is trivial, every complete metric with non-negative scalar curvature is Ricci-flat. In particular, on the connected sum of $M$ with a spin manifold, any complete metric with non-negative scalar curvature is Ricci-flat.
\end{abstract}

The study of scalar curvature on Riemannian manifolds has played a central role in differential geometry and geometric topology for several decades. A landmark result due to Gromov and Lawson \cite{GromovLawson,GromovLawsonAnnals}, using index theory, and independently to Schoen and Yau \cite{SchoenYau3,SchoenYau}, using minimal hypersurface methods, asserts that the $n$-torus does not admit any Riemannian metric of positive scalar curvature. More generally, Gromov--Lawson--Rosenberg conjectured that no aspherical manifold admits a metric of positive scalar curvature. This conjecture is known to hold for aspherical manifolds whose fundamental groups satisfy the rational strong Novikov conjecture \cite{Rosenberg,Yu}, as well as for aspherical manifolds of dimension at most five \cite{ChodoshLi,Gromov5fold}.

The non-existence of positive scalar metric holds for more generally enlargeable manifolds, introduced by Gromov--Lawson \cite{GromovLawson,GromovLawsonAnnals}, including connected sums of torus with a closed spin manifold. Using index-theoretic techniques, Su--Zhang \cite{SuZhang} established a quantitative version of this obstruction for connected sums of enlargeable manifolds with non-spin closed manifolds. Subsequently, Wang--Zhang \cite{WangZhang} and Su \cite{SuRemark} showed that the connected sum of an enlargeable manifold with any spin manifold does not admit a metric of positive scalar curvature. Related non-existence results for positive scalar curvature on non-spin connected sums, obtained via minimal hypersurface methods, can be found in \cite{ChodoshLi,ChenChuZhu,MR4291609}.

We remark that in Theorem \ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\pi_1(M)$ \cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\pi_1(M)$ is rationally injective; see Conjecture \ref{conj:Novikov} for a precise formulation. The reader may consult \cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.

\begin{theorem}\label{thm:main}
 Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
 $$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
 Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}

\begin{corollary}\label{coro:remove}
 Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
 $$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map},$$
 and the rational strong Novikov conjecture holds for $\pi_1(M)$. Then $M\backslash\Omega$ does not carry any complete metric with positive scalar curvature.
\end{corollary}

\section{Non-existence of uniformly positive scalar curvature}\label{sec:UPSC}
In this section, we prove a simple version of Theorem \ref{thm:main} to illustrate the key ideas. This version follows from classical index-theoretic tools, while the proof of the full version requires additional new ingredients and will be presented in the next section.
\begin{proposition}\label{prop:noPSC}
 Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
 $$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
 Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with uniform positive scalar curvature, i.e., scalar curvature bounded below by some positive constant.
\end{proposition}

@article{Zeidler:2019aa,
 abstract = {Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n = \sqrt{(n-1)/{n}} \cdot C$ with $C < 8(1+\sqrt{2})$ being a universal constant. This verifies a conjecture of Gromov for such manifolds. In particular, our result applies to all high-dimensional closed simply connected manifolds $M$ which do not admit a metric of positive scalar curvature. We also establish a quadratic decay estimate for the scalar curvature of complete metrics on manifolds, such as $M \times \mathbb{R}^2$, which contain $M$ as a codimension two submanifold in a suitable way. Furthermore, we introduce the "$\mathcal{KO}$-width" of a closed manifold and deduce that infinite $\mathcal{KO}$-width is an obstruction to positive scalar curvature.},
 author = {Rudolf Zeidler},
 date-added = {2021-11-10 09:44:42 -0600},
 date-modified = {2021-11-10 09:44:42 -0600},
 eprint = {1905.08520},
 journal = {Journal of Differential Geometry},
 title = {Band width estimates via the {D}irac operator},
 url = {https://arxiv.org/pdf/1905.08520.pdf},
 year = {2020},
 bdsk-url-1 = {https://arxiv.org/pdf/1905.08520.pdf}}

\begin{conjecture}[Rational strong Novikov conjecture for fundamental groups of aspherical manifolds]\label{conj:Novikov}
	Let $M$ be a closed aspherical manifold and $\Gamma=\pi_1(M)$. Then $\Gamma$ satisfies the rational strong Novikov conjecture if
	\[
	\mu\colon K_*(M)\otimes\Q \to K_*(C^*_r(\Gamma))\otimes\Q
	\]
	is injective.
\end{conjecture}

\begin{theorem}\label{thm:connectedSum}
	Let $M^n$ be a closed aspherical manifold and $N$ a spin manifold. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$.
	Then the connected sum $M\# N$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $M\# N$ is Ricci-flat.
\end{theorem}

\begin{theorem}\label{thm:main}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
	Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}

We remark that in Theorem \ref{thm:connectedSum}, the manifold $N$ is allowed to be non-compact, and the metric on $M\#N$ is not required to have bounded geometry. The rational strong Novikov conjecture for $\pi_1(M)$ \cite[Section~2]{Rosenberg} asserts that the assembly map from the $K$-homology of $M$ to the $K$-theory of the group $C^*$-algebra of $\pi_1(M)$ is rationally injective; see Conjecture \ref{conj:Novikov} for a precise formulation. The reader may consult \cite{GongWuYu} for recent progress and a survey of classes of groups known to satisfy this conjecture. To date, no counterexample to the rational strong Novikov conjecture is known.

We further generalize Theorem \ref{thm:connectedSum} to spin surgeries on aspherical manifolds.

\begin{theorem}\label{thm:main}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
	Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}

Theorem \ref{thm:connectedSum} is a special case of Theorem \ref{thm:main} by choosing $\Omega$ to be a small ball in $M$. The assumption that $\pi_1(\Omega)$ maps trivially into $\pi_1(M)$ is essential; in Example \ref{example} we construct a spin surgery on a closed aspherical manifold that admits a metric of positive scalar curvature when this condition fails.

Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \cite[Theorem~A]{XieYu14}. Inspired by the approaches in \cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \cite{WangZhu}.

As an immediate application, by taking $N=\partial\Omega\times[0,1)$ in Theorem \ref{thm:main}, we obtain the following corollary.

\begin{corollary}\label{coro:remove}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map},$$
	and the rational strong Novikov conjecture holds for $\pi_1(M)$. Then $M\backslash\Omega$ does not carry any complete metric with positive scalar curvature.
\end{corollary}

The paper is organized as follows. In Section \ref{sec:preliminary}, we review the higher indices and the strong Novikov conjecture. In Section \ref{sec:UPSC}, we prove a simplified version of Theorem \ref{thm:main}, establishing the non-existence of uniformly positive scalar curvature metrics on spin surgeries. The full proof of Theorem \ref{thm:main} is given in Section \ref{sec:nnsc}.

\medskip
\noindent\textbf{Acknowledgments.}
The author would like to thank Zhizhang Xie for helpful discussions.

Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu \cite[Theorem~A]{XieYu14}. Inspired by the approaches in \cite{SuZhang,WangXieLinf}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in \cite{WangZhu}. (Theorem~\ref{thm:connectedSum} is treated as a special case of Theorem~\ref{thm:main} by choosing $\Omega$ to be a small ball in $M$.)

Our proofs rely on Dirac operator methods. A fundamental ingredient is the relative index theorem for higher indices due to Xie and Yu (Theorem~A in Xie--Yu, \emph{[title not provided in ref\_dict]} (2014)). Inspired by the approaches in Su--Zhang, \emph{[title not provided in ref\_dict]}, and Wang--Xie--Linf, \emph{[title not provided in ref\_dict]}, we construct an explicit representative of the relative higher index and compute it in the $K$-theory of the group $C^*$-algebra. We also provide a Dirac-operator-based proof of the Ricci-flatness statement, following ideas similar to those in Wang--Zhu, \emph{[title not provided in ref\_dict]}. (In establishing the main theorem, we treat it as a special case of
\begin{theorem}\label{thm:main}
	Let $M^n$ be a closed aspherical manifold and $\Omega\subset M$ a smooth region. Assume that
	$$\pi_1(\Omega)\to \pi_1(M)\textup{ is the trivial map}.$$
	Let $N^n$ be a spin manifold with boundary, and $i\colon \partial N\to \partial \Omega$ a diffeomorphism preserving the spin structure\footnote{Although $M$ itself need not be spin, the triviality of $\pi_1(\Omega)\to\pi_1(M)$ implies that $\Omega$ and $\partial\Omega$ inherit a canonical spin structure from the spin universal cover $\widetilde M$.}. Set $Z\coloneqq (M\backslash\Omega)\cup_{\partial\Omega} N$. If the rational strong Novikov conjecture holds for $\pi_1(M)$, then $Z$ does not carry any complete metric with positive scalar curvature. Furthermore, any complete metric with non-negative scalar curvature on $Z$ is Ricci-flat.
\end{theorem}
by choosing $\Omega$ to be a small ball in $M$.)

Question:
Let $M^n$ be a closed aspherical manifold, let $N$ be a spin manifold of the same dimension $n$, and write $M\#N$ for their connected sum. Assume that the rational strong Novikov conjecture holds for $\pi_1(M)$, meaning that for $\Gamma=\pi_1(M)$ the assembly map
\[
\mu\colon K_*(M)\otimes\mathbb{Q}\to K_*(C_r^*(\Gamma))\otimes\mathbb{Q}
\]
is injective. Which of the following conclusions about complete Riemannian metrics on $M\#N$ is valid under these hypotheses?

The manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\#N$ has non-negative scalar curvature everywhere, then that metric is Ricci-flat.

Choices:
B. The manifold $M\#N$ admits no complete Riemannian metric with non-negative scalar curvature. In particular, every complete metric on $M\#N$ must have scalar curvature negative somewhere.
C. The manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature.
D. The manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature. Moreover, if a complete Riemannian metric on $M\#N$ has non-negative scalar curvature everywhere, then that metric is flat.
E. If the manifold $M\#N$ admits no complete Riemannian metric with positive scalar curvature, then the rational strong Novikov conjecture must hold for $\pi_1(M)$. Moreover, any complete Riemannian metric on $M\#N$ with non-negative scalar curvature is Ricci-flat.


\begin{prop}\label{pro ortho}
		If  Eq. \eqref{eq-k} is solvable, then $f$ satisfies

		\begin{eqnarray}\label{ortho-condition}
    \int_{\C_{\theta}}f(\xi)\<\xi, E_{\alpha}\>d\H^{n}=0,\quad \forall~ 1\leq \alpha\leq n.
\end{eqnarray}
	\end{prop}

\begin{thm}\label{main theorem}
  Let $\theta\in (0,\frac{\pi}{2}]$ and $1\leq k\leq n-1$.  Suppose $f\in C^{2}(\C_{\theta})$ is connected to $1$ in $\mathcal{L}_{-\frac{1}{k}}$.  Then there exists a $C^{3, \gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma$ in $ \ov{{\RR}^{n+1}_{+}}$, such that $\widehat{\S}$ satisfies the prescribed  $k$-th capillary area measure problem.
  Moreover, $\S$ is unique up to a horizontal translation in $\ov{{\RR}^{n+1}_{+}}$.
\end{thm}

\begin{eqnarray}\label{measure}
    S_{k}^c(\widehat{\Sigma}, \beta)\coloneqq \binom{n}{k}^{-1}\int_{\beta}(\sin^{2}\theta+\cos\theta\<\xi, e\>)\sigma_{k}(r)d{\H}^{n},\quad \text{for}~0\leq k\leq n.
\end{eqnarray}

\begin{prop}\label{pro-2.6}
 The capillary  Christoffel-Minkowski problem 		is equivalent to finding a strictly convex solution to  Eq.  \eqref{eq-k}.\end{prop}

\begin{defn}\label{def1.1}
  We say that a function $f\in C^{2}(\C_{\theta})$ is (strictly) convex if the matrix $\n^{2}f+f\sigma$ is positive semi-definite (respectively, positive definite) on $\C_{\theta}$.
\end{defn}

In convex geometry, the prescribed $k$-th area measure problem is a well-known problem. It asks whether, for a given positive function $f$ on $\mathbb{S}^{n}$, one can find a closed, strongly convex hypersurface $\Sigma$ with $k$-th symmetric function of the principal curvature radii equal to $f$ as a function on its normals? When $k=n$, it corresponds to the famous Minkowski problem, which has been fully solved through the works of Minkowski \cite{Min}, Alexandrov \cite{Alex}, Lewy \cite{Lewy}, Nirenberg \cite{Nire}, Pogorelov \cite{Pog52, Pog}, Cheng-Yau \cite{CY76} and many others. Analytically, the Minkowski problem is equivalent to solving a Monge-Amp\`ere equation on $\mathbb{S}^{n}$. The study of the Minkowski problem has played a significant role in advancing research on fully nonlinear equations, especially on the Monge-Amp\`ere equation. We can also see a closely related work by Xia \cite{Xia} for an anisotropic Minkowski problem for closed hypersurfaces.
When $k=1$, it is known as the Christoffel problem, which is equivalent to finding convex solutions to the Laplace equation on $\mathbb{S}^{n}$.  Firey \cite{Firey, Firey1} and Berg \cite{Berg}  found necessary and sufficient conditions using the Green function of the Laplacian on $\mathbb{S}^{n}$. More recently, Li-Wan-Wang \cite{LWW} expressed the Christoffel problem as a Laplace equation on the entire space $\mathbb{R}^{n+1}$ and provided a new and simpler necessary and sufficient condition for the Christoffel problem.  Further details on Christoffel's problem can be found in \cite{Chri,  Hilbert, Hur, Suss}. For the intermediate cases, i.e., $2\leq k\leq n-1$, it is referred to as the Christoffel-Minkowski problem. Guan-Ma \cite{GM}  solved this problem under a continuous homotopy condition, which was removed later by  Sheng-Wang-Trudinger \cite{STW}. In addition to the elliptic method, Chou-Wang \cite{CW00} and Bryan-Ivaki-Scheuer \cite{BIS2023} provided a flow method to solve the above problems.
For a more comprehensive description of the prescribed area measure problem in convex geometry, see \cite{Guan-note} and \cite{Sch}.

\subsection{Motivation and set up}
In recent years, there has been growing interest in the study of capillary hypersurfaces, including free boundary hypersurfaces, along with their geometry and topology, see, for instance, \cite{FL14, FS16, JWXZ, MWW-GCF, SWX, WW20, WWX22, WX2019, WeX21} and references therein.
Capillary hypersurfaces originated from physics through the work of Thomas Young. Later,
Young, Liouville, and Gauss formulated it as a variational problem using the so-called wetting energy.  For capillary hypersurfaces, we refer to the elegant book by Finn \cite{Finn}, and also a book by Maggi  \cite{Maggi}. In this paper, with a slight abuse of terminology,
a smooth compact embedded hypersurface in $\overline{{\mathbb{R}}_+^{n+1}}$  whose boundary lies on $\partial\overline{{\mathbb{R}}_+^{n+1}}$ is called {\it capillary hypersurface} if
it intersects $\partial\overline{{{\mathbb{R}}}_+^{n+1}}$ at a constant contact angle $\theta\in (0,\pi)$.
   Let $\nu$ be the unit outward normal of $\Sigma$. The contact angle $\theta $ is defined by $$ \cos (\pi -\theta)=\langle \nu, e \rangle,\quad \rm{along~\partial\Sigma},$$
	where $e\coloneqq -E_{n+1}$, $E_{n+1}$ is the $(n+1)$-th unit coordinate vector in $\overline{\mathbb{R}^{n+1}_+}$ and thus $e$ is the unit outward normal of $\partial \overline{\mathbb{R}^{n+1}_+}$.
This means that a capillary hypersurface in this paper is not necessarily a hypersurface of constant mean curvature.
If $\Sigma$ is strictly convex, then it is clear that the image of the Gauss map, namely,  $\nu$, lies on the spherical cap
\begin{eqnarray*}
		\mathbb{S}^n_\theta \coloneqq  \left\{ x\in \mathbb{S}^n \,|\,  \langle x, E_{n+1} \rangle \geq \cos \theta \right\}.
\end{eqnarray*}
Instead of using the usual Gauss map $\nu$, it is more convenient to use the following map
$$\tilde \nu\coloneqq  T \circ \nu: \Sigma \to \C_\theta,$$
where $\C_\theta$ is a spherical cap, defined by
\begin{eqnarray*}
		\C_{\theta }  \coloneqq  \left\{\xi\in \overline{\mathbb{R}^{n+1}_+} \mid |\xi-\cos\theta e|= 1 \right\},
\end{eqnarray*}
	and $T:\mathbb{S}^n_\theta\to\C_\theta$ is defined by $ T(y)\coloneqq y+\cos\theta e$,  which is a translation in the vertical direction.
 We refer to $\tilde \nu$  as the {\it capillary Gauss map} of $\Sigma$.  It turns out that $\tilde \nu: \Sigma\to \C_\theta$ is a diffeomorphism. Consequently,  one can reparametrize $\Sigma$ using its inverse on $\C_\theta$, see, e.g., \cite[Section 2]{MWWX}.

In \cite{MWW-AIM}, the authors formulated a capillary Minkowski problem, which concerns the existence of a strictly convex capillary hypersurface $\Sigma\subset \overline{\mathbb{R}^{n+1}_{+}}$ with a prescribed Gauss-Kronecker curvature on $\C_{\theta}$. We have established the following result in \cite[Theorem 1.1]{MWW-AIM}.

\

\noindent{\bf Theorem A}.
{\it Let $\theta\in (0,\frac{\pi}{2}]$ and $f\in C^2(\C_\theta)$ be a positive function  satisfying  \eqref{ortho-condition1} below.  Then there exists a $C^{3,\gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma\subset \overline{\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\tilde \nu^{-1}(\xi))=
		{f^{-1}(\xi)},$$ for all $\xi\in \C_\theta$. Moreover, $\Sigma$ is unique up to a horizontal translation in $\overline{\mathbb{R}^{n+1}_+}$.}

  \

The capillary Minkowski problem can also be viewed as a prescribed capillary area measure problem, finding a strictly convex capillary hypersurface such that the induced capillary area measure equals a prescribed Borel measure on $\C_{\theta}$. See \cite{MWW-AIM}. The area measures are an important subject in the study of convex bodies, and they are the local versions of quermassintegrals in the Brunn-Minkowski theory.  For more details about the classical Christoffel-Minkowski problem, see \cite[Chapter~8]{Sch}.

The primary objective of this article is to extend our work presented in \cite{MWW-AIM} and study the capillary Christoffel-Minkowski problem or the prescribed  $k$-th capillary area measure problem for the convex body with a capillary boundary in $\overline{\mathbb{R}^{n+1}_{+}}$.
 Firstly, we introduce a novel concept of local parallel sets for the capillary convex bodies. We say $\widehat{\Sigma}$ is a \textit{capillary convex body} (a compact convex set with non-empty interior) in $\mathbb{R}^{n+1}$ if it is a bounded closed region in $\overline{\mathbb{R}^{n+1}_+}$ enclosed by a strictly convex capillary hypersurface $\Sigma$ and  $\partial \overline{\mathbb{R}^{n+1}_{+}}$. The class of capillary convex bodies in $\overline{{\mathbb{R}}^{n+1}_+}$ is denoted by $\K_{\theta}$.   Given a Borel set $\beta\subset \C_{\theta}$ and a capillary convex body $\widehat{\Sigma}\subset \K_{\theta}$, we consider the following set :
\begin{eqnarray*}
    B_{s}(\widehat{\Sigma}, \beta)\coloneqq  \left\{Y\in \overline{{\mathbb{R}}^{n+1}_{+}} \mid  Y=X+t\tilde{\nu}(X),~0<t<s,~\text{for}~X\in \Sigma ~ \text{and} ~\tilde{\nu}(X)\in \beta \right\}.
\end{eqnarray*}
When $\beta=\C_{\theta}$, the set
\begin{eqnarray*}
    A_{s}(\widehat{\Sigma}, \C_{\theta}) \coloneqq \left\{Y\in \overline{{\mathbb{R}}^{n+1}_{+}} \mid  Y=X+s\tilde{\nu}(X), ~X\in \Sigma~\text{and}~s>0 \right\}
\end{eqnarray*}
is also a strictly convex capillary hypersurface in $\overline{{\mathbb{R}}^{n+1}_{+}}$ (see, for instance, \cite[Remark 2.17]{MWWX} or \cite[Proposition~2.3]{JWXZ}).

Direct calculation yields the following local Steiner-type formula (see also \cite[Remark~2.17]{MWWX} for a global version),
\begin{eqnarray*}
    \text{Vol}(B_{s}(\widehat{\Sigma}, \beta))&=&\sum\limits_{k=0}^{n}\frac{s^{n+1-k}}{n+1-k}\int_{\Sigma \cap \tilde{\nu}^{-1}(\beta)}(1+\cos\theta \<\nu, e\>)\sigma_{n-k}(\kappa)d\mu\notag\\
    &=&\frac{1}{n+1}\sum\limits_{k=0}^{n}s^{n+1-k}\binom{n+1}{k}S_{k}^c(\widehat{\Sigma}, \beta),
\end{eqnarray*}
where $S_k^c(\widehat{\Sigma},\beta)$ is given by
\begin{eqnarray}\label{measure}
    S_{k}^c(\widehat{\Sigma}, \beta)\coloneqq \binom{n}{k}^{-1}\int_{\beta}(\sin^{2}\theta+\cos\theta\<\xi, e\>)\sigma_{k}(r)d{\mathcal{H}}^{n},\quad \text{for}~0\leq k\leq n.
\end{eqnarray}
Here, $\kappa$ and $r$ are the set of the principal curvatures and principal curvature radii of $\Sigma$ respectively, $\sigma_{k}$ is the $k$-th elementary symmetric polynomial function, $d \mathcal{H}^n$ and $d\mu$ are the $n$-dimensional Hausdorff measure on $\C_\theta$
 and  $\Sigma$ respectively. The superscript $c$ indicates the presence of the \textit{capillary} effect in our notation. Moreover, $S_k^c (\widehat{\Sigma}, \C_{\theta})$ $(0\leq k\leq n)$ coincide with the quermassintegrals of convex capillary hypersurface introduced in  \cite[Eq. (1.8)]{WWX22}, see also \cite[Lemma~2.14]{MWWX}.

 For each capillary convex body $\widehat{\Sigma}$, by \eqref{measure} we introduce  a \textit{$k$-th capillary area measure} on $\C_{\theta}$, which is given  by \begin{eqnarray}\notag
     dS_k^c \coloneqq  \ell\sigma_{k}(\nabla^{2}h+h\sigma)d\mathcal{H}^{n}, ~~~0\leq k\leq n,
 \end{eqnarray}
 where $$\ell(\xi) \coloneqq \sin^{2}\theta+\cos\theta\<\xi, e\>,$$ and $h$ is the support function of $\Sigma$. In particular, when $\theta=\frac \pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\widehat{\Sigma}$, such that the  $k$-th capillary area measure is prescribed on $\C_{\theta}$. We formulate the problem as follows.

 \

\noindent	\textbf{Capillary Christoffel-Minkowski problem:} \textit{Given a positive, $C^{2}$ function $f$ on $\C_{\theta}$, when does  there exist a capillary convex body $\widehat{\Sigma}$ in $\overline{\mathbb{R}^{n+1}_+}$ such that
\begin{eqnarray*}
    dS_k^c=\ell fd\mathcal{H}^{n}~?
\end{eqnarray*}}

\

When $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \cite{MWW-AIM}.
For the intermediate case  $2\leq k\leq n-1$, we refer to this problem as the \textit{capillary Christoffel-Minkowski problem}. From Proposition~\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding  convex solutions to the following Hessian type equation with Robin boundary value condition,  \begin{eqnarray}
\label{eq-k}
	\begin{array}{rcll}\vspace{2mm}
		\sigma_{k}(\nabla^2 h+h\sigma)&=&f,& \quad  \hbox{ in }  \C_{\theta},\\
		\n_\mu h&=&\cot\theta h, &\quad \hbox{ on }  \partial \C_\theta,
	\end{array}
 \end{eqnarray}
where   $h$ is an unknown function, $\nabla h$ and $\nabla^2h$ are the gradient and the Hessian of $h$ on $\C_\theta$ with respect to the standard spherical metric $\sigma$ on $\C_{\theta}$ respectively  and $\mu$ is the unit outward normal of $\partial\C_\theta\subset \C_{\theta}$.
If Eq. \eqref{eq-k} is solvable, the function $f$  satisfies
\begin{eqnarray}\label{ortho-condition1}
    \int_{\C_{\theta}}f(\xi)\<\xi, E_{\alpha}\>d\mathcal{H}^{n}=0,\quad \forall~ 1\leq \alpha\leq n,
\end{eqnarray} (cf. Proposition \ref{pro ortho} for more details)
where  $\{E_i\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\overline{\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\neq n$. This phenomenon also occurs in the closed case, see for instance \cite{GM, GMZ, STW}. Before presenting the main result,  we introduce some definitions.
\begin{defn}\label{def1.1}
  We say that a function $f\in C^{2}(\C_{\theta})$ is (strictly) convex if the matrix $\nabla^{2}f+f\sigma$ is positive semi-definite (respectively, positive definite) on $\C_{\theta}$.
\end{defn}
\begin{defn}
  For $s\in \mathbb{R}$, let $\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\C_{\theta}$ such that $f$ satisfies \eqref{ortho-condition1} and $f^{s}$ is convex on $\C_{\theta}$, in the sense of Definition \ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\mathcal{L}_{s}$ if there exists a continuous path $q(t)\in \mathcal{L}_{s}$, $0\leq t\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\n_{\mu}q(t)\geq 0$ on $\partial\C_{\theta}$ for any $t\in[0, 1]$.
\end{defn}

It is worth noting that, compared to the definition of homotopy given in \cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \eqref{eq-k}.

\subsection {Main result}
The main theorem of this paper is as follows.

\noindent{\bf Theorem A}.
{\it Let $\theta\in (0,\frac{\pi}{2}]$ and $f\in C^2(\C_\theta)$ be a positive function satisfying \eqref{ortho-condition1} below. Then there exists a $C^{3,\gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma\subset \overline{\mathbb{R}^{n+1}_+}$ such that its Gauss-Kronecker curvature $K$ satisfying $$K(\tilde \nu^{-1}(\xi))=
 {f^{-1}(\xi)},$$ for all $\xi\in \C_\theta$. Moreover, $\Sigma$ is unique up to a horizontal translation in $\overline{\mathbb{R}^{n+1}_+}$.}

For each capillary convex body $\widehat{\Sigma}$, by \eqref{measure} we introduce a \textit{$k$-th capillary area measure} on $\C_{\theta}$, which is given by \begin{eqnarray}\notag
 dS_k^c \coloneqq \ell\sigma_{k}(\nabla^{2}h+h\sigma)d\mathcal{H}^{n}, ~~~0\leq k\leq n,
 \end{eqnarray}
 where $$\ell(\xi) \coloneqq \sin^{2}\theta+\cos\theta\<\xi, e\>,$$ and $h$ is the support function of $\Sigma$. In particular, when $\theta=\frac \pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\widehat{\Sigma}$, such that the $k$-th capillary area measure is prescribed on $\C_{\theta}$. We formulate the problem as follows.

When $k=n$, it reduces to the $n$-th capillary area measure and corresponds to the capillary Minkowski problem, which was solved by the authors in \cite{MWW-AIM}.
For the intermediate case $2\leq k\leq n-1$, we refer to this problem as the \textit{capillary Christoffel-Minkowski problem}. From Proposition~\ref{pro-2.6} below, the capillary Christoffel-Minkowski problem is equivalent to finding convex solutions to the following Hessian type equation with Robin boundary value condition, \begin{eqnarray}
\label{eq-k}
 \begin{array}{rcll}\vspace{2mm}
 \sigma_{k}(\nabla^2 h+h\sigma)&=&f,& \quad \hbox{ in } \C_{\theta},\\
 \n_\mu h&=&\cot\theta h, &\quad \hbox{ on } \partial \C_\theta,
 \end{array}
 \end{eqnarray}
where $h$ is an unknown function, $\nabla h$ and $\nabla^2h$ are the gradient and the Hessian of $h$ on $\C_\theta$ with respect to the standard spherical metric $\sigma$ on $\C_{\theta}$ respectively and $\mu$ is the unit outward normal of $\partial\C_\theta\subset \C_{\theta}$.
If Eq. \eqref{eq-k} is solvable, the function $f$ satisfies
\begin{eqnarray}\label{ortho-condition1}
 \int_{\C_{\theta}}f(\xi)\<\xi, E_{\alpha}\>d\mathcal{H}^{n}=0,\quad \forall~ 1\leq \alpha\leq n,
\end{eqnarray} (cf. Proposition \ref{pro ortho} for more details)
where $\{E_i\}_{i=1}^{n+1}$ is the standard coordinate unit vector of $\overline{\mathbb{R}^{n+1}_+}$. For the capillary Minkowski problem, the necessary condition \eqref{ortho-condition1} is also sufficient, while the situation becomes subtle for the general case $k\neq n$. This phenomenon also occurs in the closed case, see for instance \cite{GM, GMZ, STW}. Before presenting the main result, we introduce some definitions.
\begin{defn}\label{def1.1}
 We say that a function $f\in C^{2}(\C_{\theta})$ is (strictly) convex if the matrix $\nabla^{2}f+f\sigma$ is positive semi-definite (respectively, positive definite) on $\C_{\theta}$.
\end{defn}
\begin{defn}
 For $s\in \mathbb{R}$, let $\mathcal{L}_{s}$ be the set of positive $C^{2}$ functions $f$ on $\C_{\theta}$ such that $f$ satisfies \eqref{ortho-condition1} and $f^{s}$ is convex on $\C_{\theta}$, in the sense of Definition \ref{def1.1}. Moreover, we say $f$ is connected to $g$ in $\mathcal{L}_{s}$ if there exists a continuous path $q(t)\in \mathcal{L}_{s}$, $0\leq t\leq 1$, such that $q(0)=g$, $q(1)=f$, and $\n_{\mu}q(t)\geq 0$ on $\partial\C_{\theta}$ for any $t\in[0, 1]$.
\end{defn}

It is worth noting that, compared to the definition of homotopy given in \cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \eqref{eq-k}.

\subsection {Main result}
The main theorem of this paper is as follows.

For each capillary convex body $\widehat{\S}$, by \eqref{measure} we introduce a \textit{$k$-th capillary area measure} on $\C_{\theta}$, which is given by \begin{eqnarray}\notag
 dS_k^c \coloneqq \ell\sigma_{k}(\n^{2}h+h\sigma)d\mathcal{H}^{n}, ~~~0\leq k\leq n,
 \end{eqnarray}
 where $$\ell(\xi) \coloneqq \sin^{2}\theta+\cos\theta\<\xi, e\>,$$ and $h$ is the support function of $\S$. In particular, when $\theta=\frac \pi 2$, $dS_k^c$ coincides with the classical $k$-th area measure for the convex body, see for instance \cite[Section 5.1]{Sch}. The main goal of this paper is to identify a capillary convex body $\widehat{\S}$, such that the $k$-th capillary area measure is prescribed on $\C_{\theta}$. We formulate the problem as follows.

It is worth noting that, compared to the definition of homotopy given in \cite[Definition 1.1]{GM}, our setting includes an additional requirement: a Neumann boundary condition will be imposed. This condition plays a crucial role in establishing the constant rank theorem for ensuring the convex solutions of Eq. \eqref{eq-k}.

\

In this section, we establish the a priori estimates for the strictly convex solution of Eq. \eqref{eq-k}.
For any solution $h$ to Eq. \eqref{eq-k}, it is easy to verify that
\begin{eqnarray*}
 h(\xi)+\<\xi,E_\alpha\> ,\quad \text{ for all } 1\leq \a\leq n,
\end{eqnarray*} is still a solution. Up to a horizontal translation, we assume $h$ satisfies the following orthogonal condition:
\begin{eqnarray}\label{orth cond}
 \int_{\C_\theta} h(\xi) \<\xi,E_\alpha\> d\H^{n}=0, \quad \text{ for all } 1\leq \a\leq n.
\end{eqnarray}
Under the condition \eqref{orth cond}, we are able to establish the following a priori estimates.
\begin{thm}\label{priori estimate}
Let $\theta\in (0, \frac{\pi}{2})$. Suppose that $h$ is a strictly convex solution of Eq. \eqref{eq-k} and satisfies the condition \eqref{orth cond}. For any integer $m\geq 2$ and $\gamma\in(0,1)$, there exists a constant $C>0$ depending on $n, k,$ $\min\limits_{\C_{\theta}} f$ and $\|f\|_{C^{m}(\C_\theta)}$, such that
 \begin{eqnarray}\label{sch est}
 \|h\|_{C^{m+1,\gamma}(\C_\theta)}\leq C.
 \end{eqnarray}
\end{thm}

\begin{lem}\label{c0 est}
Let $\theta\in(0, \pi)$ and $h$ be a strictly convex solution of Eq. \eqref{eq-k} that satisfies \eqref{orth cond}.
Then there exists a positive constant $C$ depending on $n, \min\limits_{\C_{\theta}}f$, and $\max\limits_{\C_\theta} f$, such that
\begin{eqnarray*}
 0<\min_{\C_\theta} h\leq \max_{\C_\theta} h\leq C.
\end{eqnarray*}
\end{lem}
\begin{proof}
First of all, due to \eqref{orth cond}, by \cite[Lemma 3.1]{MWW-AIM}, we have $h>0$ and the origin lies in the interior of $\widehat{\partial\Sigma}$.
Let $R_{0}>0$ be the smallest number of $R$ such that
 \begin{eqnarray}\label{enclosed}
 \widehat \S\subset \widehat{R\C_\theta},
 \end{eqnarray}where $R\C_\theta\coloneqq \{\xi \in \ov{\RR^{n+1}_+}: |\xi-\cos\theta R e|=R\}$. It is clear that there exists $X_0\in \Sigma \cap R_0\C_{\theta } $. Denote $\widehat{X}_0\coloneqq \frac{X_0 }{R_0}\in \C_\theta$.
For any $\xi \in \C_\theta$, we have
 \begin{eqnarray}\label{lower bdd of h}
 h(\xi)&=&\max_{Y\in \S}\<\xi-\cos\theta e, Y\>\notag \\
 &\geq& \max \left\{0, \<\xi-\cos\theta e,X_0\> \right\}\notag \\
 &=& \max \left\{0, R_0\<\xi-\cos\theta e,\widehat X_0\> \right\}.
 ~~~\end{eqnarray}
Multiplying $f$ on both sides of \eqref{lower bdd of h} and integrating over $ \C_\theta$, we see
\begin{eqnarray}\label{R1}
 R_0\leq \left(\int_{\C_\theta}h(\xi)\cdot f(\xi)d \H^{n}\right) \left(\int_{\C_\theta}\max \left\{0, \<\xi-\cos\theta e,\widehat X_0\> \right \}\cdot f(\xi) d \H^{n}\right) ^{-1}.
\end{eqnarray}
Note that
\begin{eqnarray}\label{R2}
 \int_{\C_{\theta}}h(\xi)\cdot f(\xi) d \H^{n}=
\int_{\tilde{\nu}(\Sigma)}h(\xi)\cdot f(\xi)d\H^{n}=\int_{\S}\<X, \nu\>\sigma_{n-k}(\kappa)d\mu.
\end{eqnarray}
Together with the Alexandrov-Fenchel inequalities for convex capillary hypersurfaces \cite[Theorem~1.2]{MWWX} for $\theta\in(0, \pi)$ and the Minkowski formulas for capillary hypersurfaces \cite[Proposition~2.6]{WWX22}, we obtain
\begin{eqnarray}
 C_{n, k}\left(\int_{\S}\<X, \nu\>\sigma_{n-k}(\kappa)d\mu\right)^{\frac{1}{k+1}}&\leq& \left(\int_{\Sigma}\<X, \nu\>\sigma_{n-k+1}(\kappa)d\mu\right)^{\frac{1}{k}}\notag \\
 &=&\left(\frac{k}{n-k+1}\int_{\C_{\theta}}f(\xi)d\H^{n}\right)^{\frac{1}{k}}.\label{R3}
\end{eqnarray}
Together with \eqref{R1}, \eqref{R2} and \eqref{R3}, we conclude that
\begin{eqnarray}
 R_{0}
 &\leq& C_{n,k}\left(\int_{\C_{\theta}}f(\xi)d\H^{n}\right)^{\frac{k+1}{k}}\cdot\left(\inf\limits_{\eta\in \C_{\theta}}\int_{\C_\theta}\max\{0, \<\xi-\cos\theta e, \eta\>\}\cdot f(\xi) d \H^{n}\right) ^{-1}.\quad ~~~\label{upper bound of R0}
\end{eqnarray}
Using \eqref{enclosed} and \eqref{upper bound of R0}, we see
\begin{eqnarray*}
 h(\xi) =\<\xi-\cos\theta e , \tilde{\nu}^{-1}(\xi)\>\leq |\tilde{\nu}^{-1}(\xi)| <2R_0\leq C(n, \min_{\C_\theta} f,\max_{\C_\theta} f).
\end{eqnarray*}
 We complete the proof of Lemma \ref{c0 est}.
\end{proof}
\subsection{\texorpdfstring{$C^1$ }{}estimate}
 In this subsection, we get the gradient estimate for the strictly convex solution $h$ of Eq. \eqref{eq-k}. This estimate is a direct consequence of the convexity and the $C^{0}$ estimate.
\begin{lem}\label{C1 est}
 Let $\theta\in (0, \pi)$, and let $h$ be a strictly convex solution of Eq. \eqref{eq-k}. Then \begin{eqnarray}\label{gradient ps}
 \max\limits_{\C_{\theta}} |\n h|\leq C,
 \end{eqnarray}
 where the constant $C$ only depends on $\|h\|_{C^{0}(\C_{\theta})}$.
\end{lem}
\begin{proof}
 Consider the function
 \begin{eqnarray*}
 P=|\n h|^{2}+h^{2}.
 \end{eqnarray*}
Suppose that $P$ attains its maximum value at some point, say $\xi_{0}\in \C_{\theta}$. If $\xi_{0}\in \C_{\theta}\setminus \partial\C_{\theta}$, by the maximal condition, we have
\begin{eqnarray*}
 0=\n_{e_i}P=2h_{k}h_{ki}+2hh_{i},\quad \text{for}~1\leq i\leq n.
\end{eqnarray*}
Since $h$ is a strictly convex function, we have $\n h(\xi_{0})=0$. Combining with Lemma \ref{c0 est}, we conclude that \eqref{gradient ps} holds.

\section{Convexity}\label{sec-convex}
The primary goal of this section is to obtain a strictly convex solution of Eq. \eqref{eq-k}. Based on the celebrated result of Guan-Ma \cite{GM} and with suitable assumptions on the function $f$, we show that the convex solution of Eq. \eqref{eq-k} is strictly convex. To begin with, we establish the following key lemma.
\begin{lem}\label{lem-def}
 Let $\theta\in (0, \frac{\pi}{2})$ and $\tilde{F}(W) \coloneqq \sigma_{k}(W)$. Suppose that $h\in C^{2}(\C_{\theta})$ is a solution of Eq. \eqref{eq-k}, and assume that $W$ is positive semi-definite on $\C_\theta$. Suppose there exists a positive constant $C_{0}$ such that for a fixed integer $l $ $(k\leq l\leq n-1)$, $\sigma_{l}(W)\geq C_{0}$ for all $\xi\in \C_{\theta}$. If $f\in \mathcal{L}_{-\frac{1}{k}}$ and $\n_{\mu}f\geq 0$ on $\partial \C_{\theta}$, denote $$\phi\coloneqq \sigma_{l+1}(W(\xi)),$$ then there exist constants $C_{1}, C_{2}$ depending only on $n, k, C_{0}, \|h\|_{C^{3}},$ and $\|f\|_{C^{2}}$, such that $\phi$ satisfies
 \begin{eqnarray}
 & \tilde{F}^{ij}\phi_{ij}\leq C_{1}|\n \phi|+C_{2}\phi, & \quad \rm{in} ~~ \C_{\theta} ,\label{Diff-ine}\\
 &\n_{\mu} \phi\geq 0, & \quad {\rm{on}}~\partial \C_{\theta}.\label{boundary ine}
 \end{eqnarray}
 \end{lem}
 \begin{proof}
 The differential inequality \eqref{Diff-ine} follows from \cite[Lemma~4.1]{GM} and $f\in \mathcal{L}_{-\frac{1}{k}}$. We just need to prove the boundary condition \eqref{boundary ine}.

\

We conclude the introduction by outlining the strategy for solving  Eq. \eqref{eq-k}. If $\theta=\frac{\pi}{2}$, this problem can be reduced to the closed case using a reflection argument. Hence, we focus on the case $\theta<\frac \pi 2$ in proving Theorem \ref{main theorem}. We will employ the method of continuity.  We need to establish a  priori estimates for solutions to Eq. \eqref{eq-k} and show that the convexity is preserved. To establish the $C^{0}$ estimate, we modify the approach in  \cite[Lemma~3]{CY76}, \cite[Lemma 3.1]{MWW-AIM} and combine it with a general form of capillary Alexandrov-Fenchel inequalities for capillary hypersurfaces presented in \cite[Theorem~1.2]{MWWX} for $\theta\in(0, \pi)$.  Hence our $C^0$-estimate for \eqref{eq-k} holds for all contact angle $\theta\in(0, \pi)$, see Lemma \ref{c0 est}. The $C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate. In contrast to the closed case considered in \cite[Proposition 3.2]{GM}, our global $C^2$ estimate depends on the gradient estimates, due to the specific choice of test functions used to derive the boundary $C^2$ estimate. The choice of the test functions is motivated by the works of \cite{LTU, MQ} and \cite{MWW-AIM}.
For the preservation of the convexity, we apply the celebrated result of Guan-Ma \cite{GM} on the Constant Rank Theorem.
It is worth noting that the assumption $\theta\in (0, \frac{\pi}{2})$ is crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.
A similar assumption was employed in our previous works \cite{MWW-Weingarten} and \cite{MWW-Lp}, concerning the problems of convex capillary surfaces with prescribed $k$-th Weingarten curvature and prescribed capillary $L_p$-surface area measures for convex capillary bodies, respectively.  Nevertheless, we expect that the above results still hold for $\theta > \frac{\pi}{2}$.

\

\textbf{The rest of the article is structured as follows.}
	In Section \ref{sec2}, we list some properties of elementary symmetric functions and introduce some notations and basic facts about capillary hypersurfaces. Moreover, we prove the necessary condition \eqref{ortho-condition1} for the capillary Christoffel-Minkowski problem.
	In Section \ref{sec3}, we establish a priori estimates for the solution to Eq. \eqref{eq-k}. In Section \ref{sec-convex}, under certain assumptions on $f$, we investigate the convexity of the solution to Eq. \eqref{eq-k}.
 In the final section, by using the method of continuity, we prove Theorem \ref{main theorem}.
	\vspace{.2cm}

To prove Theorem~\ref{main theorem}, they first note that if $\theta=\frac{\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\theta<\frac \pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to Eq.~\eqref{eq-k} and show[ing] that the convexity is preserved.”

For the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \cite[Lemma~3]{CY76}, \cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\theta\in(0,\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \cite{LTU, MQ} and \cite{MWW-AIM}.

For convexity preservation, they “apply the celebrated result of Guan-Ma \cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\theta\in(0,\frac{\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of Theorem~\ref{main theorem}.

To prove the main theorem, they first note that if $\theta=\frac{\pi}{2}$ the problem “can be reduced to the closed case using a reflection argument,” so they “focus on the case $\theta<\frac \pi 2$.” The proof then “employ[s] the method of continuity,” which requires establishing “a priori estimates for solutions to
\begin{eqnarray}
\label{eq-k}
	\begin{array}{rcll}\vspace{2mm}
		\sigma_{k}(\n^2 h+h\sigma)&=&f,& \quad  \hbox{ in }  \C_{\theta},\\
		\n_\mu h&=&\cot\theta h, &\quad \hbox{ on }  \p \C_\theta,
	\end{array}
 \end{eqnarray}
 and show[ing] that the convexity is preserved.

For the estimates: the $C^{0}$ estimate is obtained by “modify[ing] the approach in \cite[Lemma~3]{CY76}, \cite[Lemma 3.1]{MWW-AIM} and combin[ing] it with a general form of capillary Alexandrov-Fenchel inequalities… in \cite[Theorem~1.2]{MWWX},” yielding a $C^0$ bound “for all contact angle $\theta\in(0,\pi)$.” The “$C^1$ estimate follows from the strict convexity of $h$ and the $C^0$ estimate.” The “global $C^2$ estimate depends on the gradient estimates,” due to the “specific choice of test functions used to derive the boundary $C^2$ estimate,” motivated by \cite{LTU, MQ} and \cite{MWW-AIM}.

For convexity preservation, they “apply the celebrated result of Guan-Ma \cite{GM} on the Constant Rank Theorem,” and emphasize that the assumption $\theta\in(0,\frac{\pi}{2})$ is “crucial not only for deriving the $C^2$-estimate, but also for showing the preservation of the convexity.” Finally, “by using the method of continuity,” they conclude the existence/uniqueness statement of the main theorem.

\label{main theorem}
  Let $\theta\in (0,\frac{\pi}{2}]$ and $1\leq k\leq n-1$.  Suppose $f\in C^{2}(\C_{\theta})$ is connected to $1$ in $\mathcal{L}_{-\frac{1}{k}}$.  Then there exists a $C^{3, \gamma}$ $(\gamma\in(0,1))$ strictly convex capillary hypersurface $\Sigma$ in $ \overline{{\mathbb{R}}^{n+1}_{+}}$, such that $\widehat{\Sigma}$ satisfies the prescribed  $k$-th capillary area measure problem.
  Moreover, $\Sigma$ is unique up to a horizontal translation in $\overline{{\mathbb{R}}^{n+1}_{+}}$.,

Question:
Let \(\theta\in(0,\pi/2]\), \(1\le k\le n-1\), \(e=E_{n+1}\), and let \(\mathcal C_\theta\subset \mathbb S^n\) be the spherical cap \(\{\xi\in\mathbb S^n:\langle \xi,e\rangle>-\cos\theta\}\). For \(s\in\mathbb R\), let \(\mathcal L_s\) be the set of positive \(C^2\)-functions \(g\) on \(\mathcal C_\theta\) such that
\[
\int_{\mathcal C_\theta} g(\xi)\langle \xi,E_\alpha\rangle\,d\mathcal H^n=0\quad\text{for }\alpha=1,\dots,n,
\]
and \(g^s\) is convex in the sense that \(\nabla^2(g^s)+g^s\sigma\) is positive semidefinite on \(\mathcal C_\theta\). Say that \(f\) is connected to \(1\) in \(\mathcal L_{-1/k}\) if there is a continuous path \(q(t)\in \mathcal L_{-1/k}\), \(0\le t\le1\), with \(q(0)=1\), \(q(1)=f\), and \(\partial_\mu q(t)\ge0\) on \(\partial\mathcal C_\theta\) for all \(t\). For a capillary convex body \(\widehat\Sigma\), its \(k\)-th capillary area measure on \(\mathcal C_\theta\) is
\[
dS_k^c=\ell(\xi)\,\sigma_k(\nabla^2h+h\sigma)\,d\mathcal H^n,
\qquad \ell(\xi)=\sin^2\theta+\cos\theta\langle \xi,e\rangle,
\]
where \(h\) is the support function of \(\Sigma\). If \(f\in C^2(\mathcal C_\theta)\) is connected to \(1\) in \(\mathcal L_{-1/k}\), which statement holds?

There exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
Moreover, any two such hypersurfaces differ only by a horizontal translation in \(\overline{\mathbb R^{n+1}_+}\).

Choices:
B.
There exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
Moreover, such a hypersurface is unique in \(\overline{\mathbb R^{n+1}_+}\) without allowing any horizontal translation.
C.
There exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
D.
For every \(\theta\in(0,\pi)\), if \(f\in C^2(\mathcal C_\theta)\) is connected to \(1\) in \(\mathcal L_{-1/k}\), then there exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta,
\]
and any two such hypersurfaces differ only by a horizontal translation in \(\overline{\mathbb R^{n+1}_+}\).
E.
If \(f\in C^2(\mathcal C_\theta)\) satisfies the orthogonality condition
\[
\int_{\mathcal C_\theta} f(\xi)\langle \xi,E_\alpha\rangle\,d\mathcal H^n=0\quad\text{for }\alpha=1,\dots,n,
\]
then there exists a \(C^{3,\gamma}\) hypersurface \(\Sigma\subset\overline{\mathbb R^{n+1}_+}\), for some \(\gamma\in(0,1)\), which is strictly convex and capillary, such that its associated capillary convex body \(\widehat\Sigma\) has prescribed \(k\)-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
Moreover, any two such hypersurfaces differ only by a horizontal translation in \(\overline{\mathbb R^{n+1}_+}\).

homotopy/convexity-preserving connectedness in \(\mathcal L_{-1/k}\)


Question:
Let $\theta\in(0,\frac\pi2]$ and $1\le k\le n-1$. Let $\mathcal C_\theta$ be the spherical cap domain for the capillary problem, with standard spherical metric $\sigma$ and outward unit normal $\mu$ along $\partial\mathcal C_\theta$. For $s\in\mathbb R$, let $\mathcal L_s$ be the set of positive $C^2$ functions $g$ on $\mathcal C_\theta$ such that
\[
\int_{\mathcal C_\theta} g(\xi)\langle \xi,E_\alpha\rangle\,d\mathcal H^n=0\qquad(1\le \alpha\le n),
\]
and $g^s$ is convex in the sense that $\nabla^2(g^s)+g^s\sigma$ is positive semidefinite on $\mathcal C_\theta$. Say that $f$ is connected to $1$ in $\mathcal L_{-1/k}$ if there exists a continuous path $q(t)\in\mathcal L_{-1/k}$, $0\le t\le1$, with $q(0)=1$, $q(1)=f$, and $\partial_\mu q(t)\ge0$ on $\partial\mathcal C_\theta$ for all $t$. For a capillary convex body $\widehat\Sigma$ with support function $h$, its $k$-th capillary area measure is
\[
dS_k^c=\ell(\xi)\,\sigma_k(\nabla^2h+h\sigma)\,d\mathcal H^n,
\qquad \ell(\xi)=\sin^2\theta+\cos\theta\langle \xi,e\rangle.
\]
A capillary hypersurface in $\overline{\mathbb R^{n+1}_+}$ is one meeting the boundary hyperplane with contact angle $\theta$. Under these assumptions, which existence statement holds?

There exists a $C^{3,\gamma}$ strictly convex capillary hypersurface $\Sigma\subset\overline{\mathbb R^{n+1}_+}$, for some $\gamma\in(0,1)$, such that the associated capillary convex body $\widehat\Sigma$ has prescribed $k$-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta,
\]
and $\Sigma$ is unique up to a horizontal translation in $\overline{\mathbb R^{n+1}_+}$.

Choices:
B.
There exists a $C^{3,\gamma}$ strictly convex capillary hypersurface $\Sigma\subset\overline{\mathbb R^{n+1}_+}$, for some $\gamma\in(0,1)$, such that the associated capillary convex body $\widehat\Sigma$ has prescribed $k$-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta,
\]
and $\Sigma$ is unique in $\overline{\mathbb R^{n+1}_+}$.
C.
There exists a $C^{3,\gamma}$ capillary hypersurface $\Sigma\subset\overline{\mathbb R^{n+1}_+}$, for some $\gamma\in(0,1)$, such that the associated capillary convex body $\widehat\Sigma$ has prescribed $k$-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta.
\]
D.
There exists a $C^{3,\gamma}$ strictly convex capillary hypersurface $\Sigma\subset\overline{\mathbb R^{n+1}_+}$, for some $\gamma\in(0,1)$, such that the associated capillary convex body $\widehat\Sigma$ has prescribed $k$-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta,
\]
for every positive $f\in C^2(\mathcal C_\theta)$ satisfying the orthogonality condition
\[
\int_{\mathcal C_\theta} f(\xi)\langle \xi,E_\alpha\rangle\,d\mathcal H^n=0\qquad(1\le \alpha\le n),
\]
and $\Sigma$ is unique up to a horizontal translation in $\overline{\mathbb R^{n+1}_+}$.
E.
There exists a $C^{3,\gamma}$ strictly convex capillary hypersurface $\Sigma\subset\overline{\mathbb R^{n+1}_+}$, for some $\gamma\in(0,1)$, such that the associated capillary convex body $\widehat\Sigma$ has prescribed $k$-th capillary area measure
\[
dS_k^c=\ell(\xi)f(\xi)\,d\mathcal H^n\quad\text{on }\mathcal C_\theta,
\]
provided only that $f\in \mathcal L_{-1/k}$; in particular, no assumption that $f$ be connected to $1$ in $\mathcal L_{-1/k}$ is needed, and $\Sigma$ is unique up to a horizontal translation in $\overline{\mathbb R^{n+1}_+}$.