\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.

The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal the correct option or uniquely cue its exact wording. There is no direct answer leakage beyond setting up the relevant theorem.

This is largely a theorem-recall/application item: the stem gives the assumptions and asks which conclusion holds. However, it is not a pure tautology because the options include stronger, weaker, and subtly altered conclusions, so the student must distinguish the exact statement.

The item requires some precision in reasoning: one distractor is a weaker true statement, others are overstatements ('non-negative scalar curvature' obstruction, 'flat' instead of 'Ricci-flat') or logical misreadings. Still, the task is mainly identifying the exact theorem conclusion rather than generating a new argument.

The distractors are strong and mathematically plausible. They target common failure modes: confusing positive vs non-negative scalar curvature, mistaking Ricci-flat for flat, and selecting a weaker true statement instead of the strongest correct one.

A solid MCQ with good distractors and little answer leakage, but it is still primarily a theorem-recall item rather than a deeply generative reasoning question.


\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 \(\mathcal L_{-1/k}\)

The stem does not explicitly state the conclusion. It supplies hypotheses and definitions, but the exact existence/uniqueness claim must still be selected from the options.

The item is very close to theorem-recall: it asks which existence statement holds under a long list of assumptions, and the correct option is essentially the theorem’s conclusion. Still, the alternatives vary meaningfully in uniqueness, strict convexity, and hypothesis strength, so it is not a pure verbatim restatement.

Some reasoning is needed to reject overstrong or underqualified conclusions, especially regarding uniqueness up to horizontal translation and the need for the path-connectedness assumption. However, the task mainly tests recognition of the precise theorem statement rather than deeper derivation.

The distractors are plausible and mathematically meaningful: one ignores translation invariance, one gives only a weaker existence claim, and others improperly weaken the hypotheses. These reflect common failure modes in reading existence/uniqueness theorems.

A reasonably well-constructed theorem-identification MCQ with strong distractors and no direct answer leakage, but it leans more toward precise recall of a stated result than toward genuinely generative mathematical reasoning.


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?

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 \(\overline{\mathbb R^{n+1}_+}\).

The stem does not explicitly state the conclusion or uniquely reveal choice A. It gives hypotheses and asks for the resulting statement, so there is no direct answer leakage.

This is essentially a theorem-recall item: the stem presents the full hypothesis set and asks for the exact conclusion. The correct option is basically the theorem statement itself.

Some reasoning is needed to distinguish subtle variants involving uniqueness, translation invariance, parameter range, and hypothesis weakening. However, the task is still mostly recognition of the exact theorem rather than genuine derivation.

The distractors are plausible and mathematically targeted: one overstates uniqueness, one gives a weaker true statement, one alters the admissible angle range, and one drops an essential connectedness hypothesis.

A solid theorem-recognition MCQ with strong distractors, but it is largely a direct restatement of a result and only moderately tests reasoning rather than generative mathematical thinking.


{\bf Approximation of quasihyperbolic geodesics.}
In spite of all these applications, finding the explicit value of the quasihyperbolic distance is not possible
except in a few special cases.  Therefore, during the past five decades many comparison metrics  have been introduced  that are easily evaluated and, at the same time,  give either upper or lower bounds for the quasihyperbolic metric.  For a survey of these comparison metrics, see \cite{hkv} and the recent  article \cite{mo}.

The problem of finding a shortest path between two given locations is a classical problem in geodesy (see e.g. \cite[Section 13.6]{VermeerRasila00}), and it has many connections to real world questions such as route planning, logistics, robotics, aviation, engineering, and communication networks. In this paper, we investigate this problem for hyperbolic type weighted metrics (see \cite{hkv}). One of the methods to study these problems is to discretize the underlying mathematical model by using graphs. The vertices of the graph are possible intermediate locations between initial and terminal locations, whereas the edges depict all the connections between vertices. The research literature of this part of mathematical graph theory is vast due to the possible savings as a result of process speedup of algorithms.

We study this shortest path problem in a planar domain with a given metric with some
special features: in particular, this metric measures not only the distance between two points
but also their location with respect to the boundary of the domain. In the cases we study, the
existence of shortest paths between two given points, i.e. length-minimizing curves or geodesics, is known by theory. A characteristic feature of the geodesics of hyperbolic type metrics is that they have a tendency to avoid the boundary of the domain as much as possible.

\subsection{Weighted metrics}
For a domain $G \subsetneq \mathbb{R}^n, n\ge 2,$ define a continuous function $w: G \to (0,\infty),$ called {\it weight function} and the {\it weighted
length} of a rectifiable curve $\gamma$ in $ G$ by
\begin{equation}
\label{w-len}
\ell_G^w(\gamma) = \int_{\gamma} \,w(z)\, |dz|\,.
\end{equation}
If $w(x)=1$ for all $x \in G\,,$ then $\ell_G^w(\gamma)$ reduces to the Euclidean length of
$\gamma$, and if
\begin{equation} \label{qh-len}
w(z)= \frac{1}{d(z, \partial G)}\,,
\end{equation}
where $d(z, \partial G)$ is the distance of $z\in G$ from the boundary of the domain, then $\ell_G^w(\gamma)$ is the {\it quasihyperbolic
length} of $\gamma\,.$
The quasihyperbolic distance
 is defined for $x,y \in G$ as
\begin{equation}\label{qhdef}
k_G(x,y)= \inf_{\gamma} \int_{\gamma} \, \frac{|dz|}{d(z, \partial G)}\,.
\end{equation}
where the infimum is taken over all rectifiable curves $\gamma$ in $G$ with
$x,y \in \gamma.$ This metric was defined by F.W. Gehring and his coauthors
B.P. Palka \cite{gp} and B. Osgood \cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \eqref{qhdef} defines a metric and that there exists a length
minimizing curve, geodesic, joining the points $x$ and $y\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \cite{RasilaTalponen14}).

Since its introduction, the quasihyperbolic metric has become a standard tool in the study
of geometric function theory, quasiconformal mappings, and analysis on metric spaces \cite{gh, gmp, hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \cite{b} at most by a constant factor \cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.

\subsection{Weighted metrics}
For a domain $G \subsetneq \mathbb{R}^n, n\ge 2,$ define a continuous function $w: G \to (0,\infty),$ called {\it weight function} and the {\it weighted
length} of a rectifiable curve $\gamma$ in $ G$ by
\begin{equation}
\label{w-len}
\ell_G^w(\gamma) = \int_{\gamma} \,w(z)\, |dz|\,.
\end{equation}
If $w(x)=1$ for all $x \in G\,,$ then $\ell_G^w(\gamma)$ reduces to the Euclidean length of
$\gamma$, and if
\begin{equation} \label{qh-len}
w(z)= \frac{1}{d(z, \partial G)}\,,
\end{equation}
where $d(z, \partial G)$ is the distance of $z\in G$ from the boundary of the domain, then $\ell_G^w(\gamma)$ is the {\it quasihyperbolic
length} of $\gamma\,.$
The quasihyperbolic distance
 is defined for $x,y \in G$ as
\begin{equation}\label{qhdef}
k_G(x,y)= \inf_{\gamma} \int_{\gamma} \, \frac{|dz|}{d(z, \partial G)}\,.
\end{equation}
where the infimum is taken over all rectifiable curves $\gamma$ in $G$ with
$x,y \in \gamma.$ This metric was defined by F.W. Gehring and his coauthors
B.P. Palka \cite{gp} and B. Osgood \cite{go}, who also studied its applications in geometric function theory. In particular, they showed that the distance function \eqref{qhdef} defines a metric and that there exists a length
minimizing curve, geodesic, joining the points $x$ and $y\,.$ Interestingly, this is not necessarily true in more general settings like Banach spaces (see \cite{RasilaTalponen14}).

Since its introduction, the quasihyperbolic metric has become a standard tool in the study
of geometric function theory, quasiconformal mappings, and analysis on metric spaces \cite{gh, gmp, hei, hkv, vjussi}, where it is typically used as the substitute for hyperbolic metric in the cases where hyperbolic metrics cannot be defined. For half-planes and half-spaces, the quasihyperbolic metric coincides with the hyperbolic metric, and when the domain $G$ is a simply connected domain of the plane, it differs from the hyperbolic metric \cite{b} at most by a constant factor \cite{gh}. This serves as a motivation for the Gromov hyperbolic uniformization theory, introduced by Bonk, Heinonen, and Koskela in the seminal paper \cite{BonkHeinonenKoskela01}, where the question of Gromov hyperbolicity of quasihyperbolic metrics plays a central role. As far as we know, the algorithm given in this paper is the first general method for approximation of geodesics and distances in quasihyperbolic metric.

In this study, quasihyperbolic geometry is investigated from the point of view of constructive approximation.
We discretize the problem by forming a grid of points in the domain $G\,.$
This grid must be fine enough considering the mutual locations of the points
and the geometric features of the domain $G\,.$ We define a graph using
these grid points and define the distance between two neighbouring points
in a manner that is compatible with the quasihyperbolic geometry. The main
step of the process is to find the shortest path in this graph. For this purpose
we use Dijkstra's algorithm \cite{Dijkstra59}. This method applies not only for the
quasihyperbolic metric but also for many other weighted metrics for which shortest
paths joining the points are known to exist. The method enables us to approximate the
quasihyperbolic geodesics by means of polygonal curves and present figures of these approximations in several polygonal
domains. We also compare our approximate values
of the quasihyperbolic distances to approximations
obtained by other methods.

After this introduction the contents of this paper is organized as follows. In Section 2 we introduce some basic notation for the metrics we use, describe how to proceed from grid points of a domain to
a graph, and some facts about special functions. In Section 3 we discuss computing distances in graphs and introduce Dijkstra's algorithm and discuss implementation of the algorithm for approximation of geodesics based on Dijkstra's algorithm. In Section 4 we use this algorithm for approximation of hyperbolic and quasihyperbolic geodesics in the unit disk, with an aim of validating the method and understanding its accuracy. We also consider conformal mappings of the upper half plane onto two polygonal domains, expressed in terms of Schwarz-Christoffel integrals. In Section 5 we investigate cases where non-uniqueness and bifurcation of geodesics is expected from theoretical considerations. In Sections 6--8 we consider other interesting examples where hyperbolic and quasihyperbolic geodesics can be approximated by using our methods. Finally, in Section 9, we give our final conclusions.
\section{Preliminaries}
In this section, we give the formulas and results needed later in our examples. We first focus on hyperbolic geometry and then present the necessary tools in order to handle conformal mappings of the upper half plane onto a quadrilateral domain.

\begin{nonsec}{\bf Distance ratio metric.}
 A useful comparison
metric for the quasihyperbolic metric is the distance ratio metric \cite{gp}
\begin{equation}\label{jdef}
j_G(x,y)= \log \left( 1+ \frac{|x-y|}{\min\{ d(x, \partial G), d(y, \partial G)\}}\right) \,,
\end{equation}
where $x,y \in G$.
The metrics $j_G$ and $k_G$ are not conformally invariant, and they are only
similarity invariant.
If $G \in \{ \mathbb{B}^2, \mathbb{H}^2\}, $ then for
$x,y \in G$
 \begin{equation} \label{jrho}
 j_G(x,y) \le \rho_G(x,y) \le 2 j_G(x,y) \,,
 \end{equation}
 \begin{equation} \label{krho}
 k_G(x,y) \le \rho_G(x,y) \le 2 k_G(x,y) \,.
 \end{equation}
In fact, $k_G \equiv \rho_G $ for $G= \mathbb{H}^2.$
The geodesics of the quasihyperbolic metric are smooth
\cite{mar}.

In view of the above two-sided inequalities \eqref{jrho} and \eqref{krho}
and the lower bound \eqref{kjineq} it is natural to ask for which classes of
domains there is a majorant for $k_G$ in terms of $j_G.$
 Indeed, inequalities between
metrics can be used to define some important classes of domains. Thus, for instance, the class of simply connected plane domains $G$ for which there exists a constant $A \ge 1$ such that for all $x,y \in G$
\begin{equation} \label{unifdef}
k_G(x,y) \le A \, j_G(x,y)
\end{equation}
defines the class of quasidisks studied in \cite{gh}. Domains of this class, in turn, are
special cases of the so called $\varphi$-uniform domains \cite{hkv}.

\begin{nonsec}{\bf Dijkstra's algorithm compared to other path finding methods.}
Despite its age and simplicity, Dijkstra's algorithm remains one of the fastest and most used algorithms of computing distances in weighted graphs. If negative edge weights are allowed (but there are no negative cycles), the distances can be computed using the Bellman-Ford-Moore algorithm \cite{Bellman58,Ford56,Moore59}. If the digraph is unweighted, it is faster to use a simple algorithm adapted from breadth-first-search. During the preparation of this article, Duan et al. \cite{Duan25} showed that there exists an algorithm running in $\mathcal{O} (m \log^{2/3} n)$ time for computing the distances from one vertex to all others in a weighted digraph with non-negative edge weights. This is the long-awaited improvement to Dijkstra's algorithm, which has turned out difficult to improve due to the sorting step. However, Duan et al. make several assumptions to simplify the presentation of their algorithm. Most notably, the graphs are assumed to be sparse and every vertex has constant in- and out-degree. The algorithm they present is generalizable to graphs without these constraints. Dijkstra's algorithm is readily available and well-optimized for different programming languages, which is why we chose to use Dijkstra's algorithm and not implement Duan et al.'s algorithm for the purposes of our experiments.
\end{nonsec}

We discretize the problem by forming a grid of points in the domain $G$ (``fine enough'' with respect to the locations of points and the geometric features of $G$). Using these grid points, we define a graph and define the distance between two neighbouring points ``in a manner that is compatible with the quasihyperbolic geometry.'' The main step is then to find the shortest path in this graph; for this we use Dijkstra's algorithm. This yields approximations of quasihyperbolic geodesics by polygonal curves (shortest graph paths), and corresponding approximate quasihyperbolic distances, which are then compared to approximations obtained by other methods.

Question:
Let \(G\subsetneq \mathbb{R}^n\) be a domain, and define the quasihyperbolic distance for \(x,y\in G\) by
\[
k_G(x,y)=\inf_{\gamma}\int_{\gamma}\frac{|dz|}{d(z,\partial G)},
\]
where the infimum is over all rectifiable curves \(\gamma\subset G\) joining \(x\) and \(y\), and \(d(z,\partial G)\) denotes the Euclidean distance from \(z\) to the boundary of \(G\). Which statement about evaluating or bounding \(k_G\) is asserted?

Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, many comparison metrics have been introduced that are easy to evaluate and provide either upper bounds or lower bounds for the quasihyperbolic metric.

Choices:
B. For every domain \(G\subsetneq \mathbb{R}^n\), the explicit value of the quasihyperbolic distance can be found by discretizing \(G\) finely enough; thus comparison metrics are introduced mainly to recover the same values in a simpler closed form.
C. Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, comparison metrics have been introduced that are easy to evaluate and are used to estimate the quasihyperbolic metric.
D. Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, many comparison metrics have been introduced that are easy to evaluate and provide both upper bounds and lower bounds for the quasihyperbolic metric simultaneously.
E. Except in a few special cases, the explicit value of the quasihyperbolic distance cannot be found; consequently, one introduces comparison metrics that, for every pair \(x,y\in G\), converge to \(k_G(x,y)\) under arbitrarily fine graph discretizations of \(G\).

The stem defines the quasihyperbolic metric but does not explicitly or implicitly reveal the correct claim about computability or comparison metrics. The answer must be identified from the options themselves.

The question is essentially a recall/recognition task asking which textual statement is asserted, and the correct option is a near-verbatim restatement of that assertion rather than an application or consequence.

There is some pressure to distinguish among closely related claims, especially A vs. C vs. D, but the task is mostly precise paraphrase matching rather than genuine mathematical generation or derivation.

The distractors are plausible and target common confusions: exact computability via discretization, weakening 'upper or lower bounds' to vague estimation, and incorrectly strengthening to simultaneous two-sided bounds. They are distinct and mathematically aligned.

A solid recognition-style MCQ with strong distractors and no answer leakage, but it is largely tautological and tests textual recall more than generative mathematical reasoning.