\label{thm 1}
Let $\Omega\subset \mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\in C^2([0, +\infty))$ satisfies  $h'(r)\ge 0$ and $h''(r)\ge 0$ for $r\in (0, +\infty)$. Let  $B \subset \mathbb{R}^m$  be the  origin-centered  round ball with the same $\gamma_h$-volume as $\Omega$, namely $ \int_{\Omega} e^{h(|x|)}\, dx= \int_{B} e^{h(|x|)}\, dx$. Denote by $\sigma_1(B; \gamma_h)$ the first nonzero Steklov eigenvalue of \eqref{eq 1.4} on $B$, and $\l_{2,\alpha}(\Omega; \gamma_h)$ the second Robin eigenvalue of \eqref{eq 1.1} on $\Omega$.  If  $\alpha \in [-\sigma_1(B; \gamma_h), 0]$, then
\begin{align}\label{eq 1.6}
\l_{2,\alpha}(\Omega; \gamma_h) \le \l_{2,\alpha}(B; \gamma_h).
\end{align}
Equality holds  if and only if
$\Omega=B$.

\begin{theorem}\label{thm 1}
Let $\Omega\subset \R^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\in C^2([0, +\infty))$ satisfies  $h'(r)\ge 0$ and $h''(r)\ge 0$ for $r\in (0, +\infty)$. Let  $B \subset \R^m$  be the  origin-centered  round ball with the same $\gamma_h$-volume as $\Omega$, namely $ \int_{\Omega} e^{h(|x|)}\, dx= \int_{B} e^{h(|x|)}\, dx$. Denote by $\sigma_1(B; \gamma_h)$ the first nonzero Steklov eigenvalue of \eqref{eq 1.4} on $B$, and $\l_{2,\a}(\Omega; \gamma_h)$ the second Robin eigenvalue of \eqref{eq 1.1} on $\Omega$.  If  $\a \in [-\sigma_1(B; \gamma_h), 0]$, then
\begin{align}\label{eq 1.6}
\l_{2,\a}(\Omega; \gamma_h) \le \l_{2,\a}(B; \gamma_h).
\end{align}
Equality holds  if and only if
$\Omega=B$.
\end{theorem}

The classical Szeg\"o-Weinberger inequality \cite{Sz54, Wei56} states that the ball uniquely maximizes the second (first nonzero) Neumann eigenvalue of Laplacian among all bounded domains of the same volume in Euclidean space. Using the arguments of Szeg\"o and Weinberger, analogous inequalities for the second Neumann eigenvalues have been established in various settings: Ashbaugh-Benguria \cite{AB95} extended the the inequality to bounded domains in hemisphere and in hyperbolic space; Chiacchio and di Blasio \cite{CB12} obtained a sharp isopermetric inequality for the first nonzero Neumann eigenvalue for bounded domians symmetric about the origin in Gauss space.
Moreover, it is expected (cf. \cite[Problem 4.41]{Hen17}) that for a range of negative Robin parameters, the second Robin eigenvalue should be maximized by a round ball. This expectation has recently been confirmed by Freitas and Laugesen via the Szeg\"o \cite{FL20} and the Weinberger \cite{FL21} approaches.
For further results on Szeg\"o-Weinberger type inequalities, we refer to \cite{AS96, As99, BBC20, BCB16, LL23, LWW22, LWW23, MW24, Wang19, XW23} and references therein.

Eigenvalue optimization problems for the weighted Laplacian have also attracted considerable attention. For instance, Chiacchio and di Blasio \cite{CB12} proved that, among all origin-symmetric regions with fixed Gaussian volume, the ball maximizes the first nonzero Neumann eigenvalue; Brock, Francesco and di Blasio \cite{BCB16} obtained optimal Szeg\"o-Weinberger type inequalities under a weighted measure.
Additional results on weighted eigenvalue problems can be found in \cite{BCHT13, BCKT16, BCB16, CM24, CG22, GW26} and the references therein.

Inspired by the work of Brock, Francesco and di Blasio \cite{BCB16} on the weighted Laplace operator and that of Freitas and Laugesen \cite{FL21} on the second Robin eigenvalue, we study the shape optimization problem for the second Robin eigenvalue of the following class of problems
\begin{align}\label{eq 1.1}
 \begin{cases}
 -\operatorname{div} (e^{h(|x|)}\nabla u)=\lambda e^{h(|x|)} u, \quad & x\in \Omega,\\
 \frac{\partial u}{\partial \nu}+\alpha u=0, \quad & x\in \partial \Omega,
 \end{cases}
\end{align}
where $\alpha\in \mathbb{R}$. Here and in the sequel, $\Omega$ denotes a bounded connected domain in $\mathbb{R}^m$ with Lipchitz boundary, $\nu$ is the outward normal to $\partial \Omega$, and $h(r)$ is a $C^2$ function on $[0, +\infty)$. We define the weighted measure associated with $h$ by
$$
d\gamma_h=e^{h(|x|)}\,dx, \qquad x\in \mathbb{R}^m,
$$
and denote by $W^{1,2}(\Omega; \gamma_h)$ the weighted Sobolev space equipped with the norm
\begin{align*}
||u||_{W^{1,2}(\Omega;\gamma_h)}:=\left(\int_\Omega u^2 \, d\gamma_h\right)^{1/2}+ \left(\int_\Omega |\nabla u|^2 \, d\gamma_h\right)^{1/2}.
\end{align*}

It is well known that the problem \eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\l_{k, \alpha}(\Omega; \gamma_h)$ for $k=1,2, \cdots$, which satisfy
 \begin{align*}
 \l_{1, \alpha}(\Omega;\gamma_h) < \l_{2, \alpha}(\Omega;\gamma_h) \leq \l_{3, \alpha}(\Omega;\gamma_h) \leq \cdots \to +\infty,
 \end{align*}
with each eigenvalue repeated according to its multiplicity.
By standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \eqref{eq 1.1} admit the variational characterizations:
\begin{align}
 \lambda_{1,\alpha}(\Omega; \gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}\, dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}\},
\end{align}
where $dA$ is the
induced measure on $\partial \Omega$;
and
\begin{align}\label{eq 1.3}
 \lambda_{2,\alpha}(\Omega;\gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}, \int_{\Omega}uu_{1}d\gamma_h=0\},
\end{align}
where $u_1(x)$ is an eigenfunction corresponding to $\l_{1, \alpha}(\Omega;\gamma_h)$.

To state our main results, we introduce the following Steklov eigenvalue problem on $\Omega$
\begin{align}\label{eq 1.4}
\begin{cases}
 -\operatorname{div} (e^{h(|x|)}\nabla u)=0 \quad & x\in \Omega,\\
 \frac{\partial u}{\partial \nu}=\sigma u, \quad & x\in \partial \Omega.
\end{cases}
\end{align}
The first nonzero Steklov eigenvalue can be characterized variationally by
\begin{align}
\sigma_1(\Omega; \gamma_h)=\inf\left\{ \frac{\int_{\Omega} |\nabla u|^2 \, d\gamma_h}{\int_{\partial \Omega} u^2 e^{h}\, dA} : u\in W^{1,2}(\Omega; \gamma_h )\setminus\{0\},\;\; \int_{\partial \Omega} ue^{h(|x|)}\ dA=0 \right\}.
\end{align}
 In this paper, we establish the following optimal isoperimetric inequality for the second Robin eigenvalue of \eqref{eq 1.1}.

\begin{align}\label{eq 1.1}
    \begin{cases}
        -\div (e^{h(|x|)}\nabla u)=\l e^{h(|x|)} u, \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}+\a u=0, \quad & x\in \p \Omega,
    \end{cases}
\end{align}

\begin{align}\label{eq 1.4}
\begin{cases}
      -\div (e^{h(|x|)}\nabla u)=0 \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}=\sigma u, \quad & x\in \p \Omega.
\end{cases}
\end{align}

It is well known that the problem \eqref{eq 1.1} has discrete spectrum. We denote its eigenvalue by $\l_{k, \a}(\Omega; \gamma_h)$ for $k=1,2, \cdots$, which satisfy
 \begin{align*}
 \l_{1, \a}(\Omega;\gamma_h) < \l_{2, \a}(\Omega;\gamma_h) \leq \l_{3, \a}(\Omega;\gamma_h) \leq \cdots \to +\infty,
 \end{align*}
with each eigenvalue repeated according to its multiplicity.
By standard spectral theory for self-adjoint compact operators, the first two eigenvalues of \eqref{eq 1.1} admit the variational characterizations:
\begin{align}
 \lambda_{1,\alpha}(\Omega; \gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}\, dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}\},
\end{align}
where $dA$ is the
induced measure on $\p \Omega$;
and
\begin{align}\label{eq 1.3}
 \lambda_{2,\alpha}(\Omega;\gamma_h)=\inf\{\frac{\int_{\Omega}|\nabla u|^{2}d\gamma_h+\alpha\int_{\partial \Omega}u^{2} e^{h}dA}{\int_{\Omega}u^{2}d\gamma_h}: u\in W^{1,2}(\Omega;\gamma_h)\setminus\{0\}, \int_{\Omega}uu_{1}d\gamma_h=0\},
\end{align}
where $u_1(x)$ is an eigenfunction corresponding to $\l_{1, \a}(\Omega;\gamma_h)$.

By taking $\a=-\sigma_1(B;\gamma_h)$, Theorem \ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \cite[Corollary 1.2]{MZ24}.
\begin{corollary}\label{cor}
Under the hypotheses of Theorem \ref{thm 1}, we have
\begin{align*}
\sigma_1(\Omega; \gamma_h)\le \sigma_1(B; \gamma_h).
\end{align*}
Equality holds if and only if $\Omega$ is isometric to the round ball $B$.
\end{corollary}

\begin{proposition} \label{prop 2.4}
Suppose $h'(r)\ge 0$, and $\a\ge -\sigma_1(B(R); \gamma_h)$. Then
\begin{align*}
 \lambda_{2,\alpha}(B(R); \gamma_h)\geq 0.
\end{align*}
\end{proposition}
\begin{proof}
 Let $g(r)$ be as defined in Proposition \ref{prop 2.3}.
 Since
 \begin{align*}
 \int_{\partial B(R)}g(r)\frac{x_{i}}{r}e^{h(r)}dA=0, 1\leq i\leq m,
 \end{align*}
 the functions $u_{i}=g(r)\frac{x_{i}}{r}, 1\leq i\leq m,$ are admissible test functions for $\sigma_{1}(B(R), \gamma_h)$. Therefore,
 \begin{align}\label{eq 2.22}
 \sum_{i=1}^{m}\int_{B(R)}|\nabla u_{i}|^{2}\, d\gamma_h\geq \sigma_1(B(R); \gamma_h) \sum_{i=1}^{m}\int_{\partial B(R)}|u_{i}|^{2}e^{h(r)}dA,
 \end{align}
 where $dA$ is the induced measure on $\p B$.
Using
 \begin{align*}
 \sum_{i=1}^{m}|\nabla u_{i}|^{2}=g'(r)^{2}+\frac{m-1}{r^{2}}g(r)^{2}.
 \end{align*}
 and the fact $h(r)\le h(R)$ for $r\in[0, R]$, we estimate from \ref{eq 2.22}
 \begin{align}\label{eq 2.23}
 0&\leq \sum_{i=1}^{m}\int_{B(R)}|\nabla u_{i}|^{2}e^{h(r)}dx- \sigma_1(B(R); \gamma_h)\sum_{i=1}^{m}\int_{\partial B(R)}|u_{i}|^{2}e^{h(r)}dA\nonumber\\
 &\leq \omega_{m-1}\int_{0}^{R}\left(g'(r)^{2}+\frac{m-1}{r^{2}}g^{2}\right)e^{h(r)}r^{m-1}dr+\alpha \omega_{m-1} g(R)^{2}e^{h(R)}R^{m-1},
 \end{align}
 where $\omega_{m-1}$ denotes the surface area of the unit sphere $\mathbb{S}^{m-1}$ in $\R^m$.
Using integration by parts and \eqref{eq 2.11}, we compute
\begin{align*}
 &\int_{0}^{R}\left(g'(r)^{2}+\frac{m-1}{r^{2}}g^{2}\right)e^{h(r)}r^{m-1}dr\\
 =\quad &gg'e^{h(r)}r^{m-1}\Big|_{0}^{R}-\int_{0}^{R}g(g'e^{h(r)}r^{m-1})'dr+\int_{0}^{R}\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\
 =\quad &g(R)g'(R)e^{h(R)}R^{m-1}-\int_{0}^{R}g(g''+(\frac{m-1}{r}+h')g')e^{h(r)}r^{m-1}dr\\
 &+\int_{0}^{R}\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr\\
 =&g(R)g'(R)e^{h(R)}R^{m-1}+\int_{0}^{R}g^{2}(\lambda_{2,\alpha}(B(R);\gamma_h)-\frac{m-1}{r^{2}})e^{h(r)}r^{m-1}dr\\
 &+\int_{0}^{R}\frac{m-1}{r^{2}}g^{2}e^{h(r)}r^{m-1}dr.
\end{align*}
Substituting this equality into inequality \eqref{eq 2.23} yields
 \begin{align*}
 0\leq& \lambda_{2,\alpha}(B(R);\gamma_h)\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}dr+g(R)e^{h(R)}R^{m-1}(g'(R)+\alpha g(R))\\
 =&\lambda_{2,\alpha}(B(R);\gamma_h)\int_{0}^{R}g^{2}(r)e^{h(r)}r^{m-1}\, dr,
 \end{align*}
which implies $\lambda_{2,\alpha}(B(R);\gamma_h)\geq 0$.
\end{proof}

From now on, we assume $\Omega\subset \R^m$ is a bounded Lipschitz domain symmetric about the origin, and $B\subset \R^m$ is the origin-centered round bound with the same $\gamma_h$ voulme as $\Omega$. Let $R$ be the radius of $B$. We extend the function $g$ defined in \eqref{eq 2.11} by
\begin{equation}\label{eq 3.1}
g(r)=g(R) e^{-\a(r-R)}\text{\quad for \quad} r\ge R.
\end{equation}
By definition, $g$ is continuously differentiable on $(0,\infty)$. If $\a\le 0$, then by Proposition \ref{prop 2.2}, $g$ is increasing on $(0,\infty)$. In the sequel, $\sigma_1(B; \gamma_h)$ denotes the first nonzero Steklov eigenvalue of $B$ as defined in \eqref{eq 2.2}.

{\bf Case 2.} $r\ge R$. For $r\ge R$,
\begin{align*}
 F(r)=(-\alpha^{2}+\frac{m-1}{r^{2}}+\alpha\frac{m-1}{r}+\alpha h^{'})g^{2}(r),
\end{align*}
and a direct computation yields
\begin{align*}
 F'(r)=(2\alpha^{3}+\alpha h''-2\alpha^{2}h'-2\alpha^{2}\frac{m-1}{r}-3\alpha \frac{m-1}{r^{2}}-2\frac{m-1}{r^{3}})g^{2}(r).
\end{align*}
Since $\alpha\leq 0$, $h'\geq 0$ and $h^{''}\geq 0$, we obtain
\begin{align*}
 F'(r)\le& \left(-2\alpha^{2}\frac{m-1}{r}-3\alpha \frac{m-1}{r^{2}}-2\frac{m-1}{r^{3}}\right) g(r)^2\\
 =&-\frac{m-1}{r}(2(\alpha+\frac{3}{4r})^{2}+\frac{7}{8}\frac{1}{r^{2}})g(r)^2<0,
\end{align*}
proving that $F(r)$ is monotonically decreasing on $(0, R)$.
\end{proof}
 The main idea in proving Theorem \ref{thm 1} is to construct trial functions for $\l_{2, \a}$ (see \eqref{eq 1.3}) using techniques introduced by Weinberger. We first recall a monotonicity lemma for weighted symmetrization.
\begin{lemma}\label{Lemma 3.2}
 Let $\Omega\subset \R^m$ be a bounded set, and let $B$ be the origin-centered round ball with the same $\gamma_h$ volume as $\Omega$. If $f(r)$ is monotonically decreasing on $[0,+\infty)$, then
 \begin{align*}
 \int_\Omega f(|x|)\, d\gamma_m\le \int_B f(|x|)\, d\gamma_m.
 \end{align*}
 Equality holds if and only if $\Omega=B$.
\end{lemma}
\begin{proof}
 Lemma 3.2 follows directly from Hardy-Littlewood inequality; see the proof of inequality (4.24) in \cite[Page 213]{BCB16}.
\end{proof}

Now we now proceed to prove Theorem \ref{thm 1}.
\begin{proof}
 Let $\Omega$ be a bounded Lipschitz domain symmetric about the origin in $\R^m$, and let $B\subset \R^m$ be the origin-centered round ball of radius $R$ satisfying
 $$
\int_B \, d\gamma_h =\int_\Omega \, d\gamma_h.
 $$
Define $g(r)$ for $r\in [0, +\infty)$ as in \eqref{eq 3.1}, and for each $1\le i\le m$ set
\begin{align*}
 v_i(x)=g(|x|)\frac{x_i}{|x|},
\end{align*}
where $x=(x_1, x_2, \cdots, x_m)\in \R^m$.

By taking $\alpha=0$, Theorem \ref{thm 1} implies that the ball centered at the
origin is the unique set maximizer the second Neumann eigenvalue $\mu_2(\Omega;\gamma_h)$ among all bounded Lipschitz domains $\Omega$ in $\mathbb{R}^m$ with prescribed $\gamma_h$-measure and symmetric about the origin. This result was previously  proved by Brock, Chiacchio, and di Blasio (Theorem 1.2 of \cite{BCB16}). Note that Theorem \ref{thm 1} applies in particular  to $\l_{2, \alpha}(\Omega; e^{|x|^2/2})$. Hence, our result provides information about  eigenvalues of the problem
$$
-\Delta u-x\cdot \nabla u=\lambda u
$$
which has been widely studied in literature (see, e.g. \cite{BCM12, BMP13}).
The symmetry assumption on $\Omega$ in Theorem \ref{thm 1}
 ensures the validity of  the orthogonality conditions \eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is  radially
 symmetric about the origin.
It remains an interesting question that whether \eqref{eq 1.6}
holds in the case of the Gauss measure $e^{-|x|^2/2}dx$.

By taking $\alpha=-\sigma_1(B;\gamma_h)$, Theorem \ref{thm 1} yields the following Brock–Weinstock inequality, which was proved by Brock and Chiacchio \cite[Theorems 1.4 and 1.5]{BC25} and by Mao and Zhang \cite[Corollary 1.2]{MZ24}.
\begin{corollary}\label{cor}
Under the hypotheses of Theorem \ref{thm 1}, we have
\begin{align*}
\sigma_1(\Omega; \gamma_h)\le \sigma_1(B; \gamma_h).
\end{align*}
Equality holds if and only if $\Omega$ is isometric to the round ball $B$.
\end{corollary}

The main technique in the proof of Theorem \ref{thm 1} relies on Weinberger’s trick, in which  test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties in the proof include analyzing the properties of the second eigenfunction of \eqref{eq 1.1} on the round ball  and establishing the required monotonicity to apply Weinberger’s method. The rest of the paper is organized as follows.
In Section \ref{sect2}, we study properties of the Robin and Steklov eigenvalue  problems for balls in the weighted space. In Section  \ref{sect3}, we prove Theorem \ref{thm 1} and Corollary \ref{cor}.

The proof of Theorem~\ref{thm 1} "relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues." The "main difficulties" are "analyzing the properties of the second eigenfunction of \eqref{eq 1.1} on the round ball" and "establishing the required monotonicity to apply Weinberger’s method." The symmetry assumption on $\Omega$ "ensures the validity of the orthogonality conditions \eqref{eq 3.5}, as the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin."

The proof of the main theorem relies on Weinberger’s trick, in which test functions are constructed using eigenfunctions on a round ball to estimate eigenvalues. The main difficulties are analyzing the properties of the second eigenfunction of
\begin{align}\label{eq 1.1}
    \begin{cases}
        -\div (e^{h(|x|)}\nabla u)=\l e^{h(|x|)} u, \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}+\a u=0, \quad & x\in \p \Omega,
    \end{cases}
\end{align}
on the round ball and establishing the required monotonicity to apply Weinberger’s method. The symmetry assumption on $\Omega$ ensures the validity of the orthogonality conditions
\begin{align}\label{eq 3.5}
        \int_\Omega v_i(x)u(x)   \, d\gamma_h=0, \quad i=1,2,\cdots, m.
    \end{align}
as the weighted measure $e^{h(|x|)}$ is radially symmetric about the origin.

\label{thm 1}
Let $\Omega\subset \mathbb{R}^m$ be a bounded  Lipschitz domain symmetric about the origin. Suppose $h(r)\in C^2([0, +\infty))$ satisfies  $h'(r)\ge 0$ and $h''(r)\ge 0$ for $r\in (0, +\infty)$. Let  $B \subset \mathbb{R}^m$  be the  origin-centered  round ball with the same $\gamma_h$-volume as $\Omega$, namely $ \int_{\Omega} e^{h(|x|)}\, dx= \int_{B} e^{h(|x|)}\, dx$. Denote by $\sigma_1(B; \gamma_h)$ the first nonzero Steklov eigenvalue of
\begin{align}\label{eq 1.4}
\begin{cases}
      -\div (e^{h(|x|)}\nabla u)=0 \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}=\sigma u, \quad & x\in \p \Omega.
\end{cases}
\end{align}
on $B$, and $\l_{2,\alpha}(\Omega; \gamma_h)$ the second Robin eigenvalue of
\begin{align}\label{eq 1.1}
    \begin{cases}
        -\div (e^{h(|x|)}\nabla u)=\l e^{h(|x|)} u, \quad & x\in \Omega,\\
        \frac{\p u}{\p \nu}+\a u=0, \quad & x\in \p \Omega,
    \end{cases}
\end{align}
on $\Omega$.  If  $\alpha \in [-\sigma_1(B; \gamma_h), 0]$, then
\begin{align}\label{eq 1.6}
\l_{2,\alpha}(\Omega; \gamma_h) \le \l_{2,\alpha}(B; \gamma_h).
\end{align}
Equality holds  if and only if
$\Omega=B$.,

Question: Let Ω ⊂ R^m be a bounded Lipschitz domain symmetric about the origin, and let h ∈ C^2([0,∞)) satisfy h'(r) ≥ 0 and h''(r) ≥ 0 for all r > 0. Define the weighted measure dγ_h = e^{h(|x|)} dx. Let B ⊂ R^m be the origin-centered round ball with the same γ_h-volume as Ω, i.e. ∫_Ω e^{h(|x|)} dx = ∫_B e^{h(|x|)} dx. For a domain D, let λ_{2,α}(D; γ_h) denote the second eigenvalue of the weighted Robin problem -div(e^{h(|x|)}∇u) = λ e^{h(|x|)}u in D, with boundary condition ∂u/∂ν + αu = 0 on ∂D. Let σ_1(B; γ_h) denote the first nonzero Steklov eigenvalue on B for -div(e^{h(|x|)}∇u) = 0 in B, with ∂u/∂ν = σu on ∂B. If α ∈ [-σ_1(B; γ_h), 0], which quantitative estimate holds?

One has λ_{2,α}(Ω; γ_h) ≤ λ_{2,α}(B; γ_h), and equality holds if and only if Ω = B.

Choices:
B. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\) for every \(\alpha \le 0\), and equality holds if and only if \(\Omega = B\).
C. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\).
D. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \le \lambda_{2,\alpha}(B; \gamma_h)\) whenever \(\alpha \in [-\sigma_1(\Omega; \gamma_h),0]\), and equality holds if and only if \(\Omega = B\).
E. One has \(\lambda_{2,\alpha}(\Omega; \gamma_h) \ge \lambda_{2,\alpha}(B; \gamma_h)\), and equality holds if and only if \(\Omega = B\).

sharp range for \alpha tied to monotonicity/Steklov threshold

dependence of the threshold on B rather than on \Omega

extremal direction from Weinberger trial-function argument


Question:
Let \(\Omega\subset \mathbb{R}^m\) be a bounded Lipschitz domain that is symmetric about the origin, and let \(h\in C^2([0,\infty))\) satisfy \(h'(r)\ge 0\) and \(h''(r)\ge 0\) for all \(r>0\). Define the weighted measure \(d\gamma_h=e^{h(|x|)}\,dx\). Let \(B\subset \mathbb{R}^m\) be the origin-centered round ball such that
\[
\int_{\Omega} e^{h(|x|)}\,dx=\int_B e^{h(|x|)}\,dx.
\]
For a domain \(D\), let \(\lambda_{2,\alpha}(D;\gamma_h)\) be the second eigenvalue of the weighted Robin problem
\[
\begin{cases}
-\operatorname{div}(e^{h(|x|)}\nabla u)=\lambda e^{h(|x|)}u, & x\in D,\\
\dfrac{\partial u}{\partial \nu}+\alpha u=0, & x\in \partial D,
\end{cases}
\]
and let \(\sigma_1(B;\gamma_h)\) be the first nonzero Steklov eigenvalue on \(B\) for
\[
\begin{cases}
-\operatorname{div}(e^{h(|x|)}\nabla u)=0, & x\in B,\\
\dfrac{\partial u}{\partial \nu}=\sigma u, & x\in \partial B.
\end{cases}
\]
Assume that \(\alpha\in[-\sigma_1(B;\gamma_h),0]\). Which conclusion about the second Robin eigenvalue is valid under these hypotheses?

\(\lambda_{2,\alpha}(\Omega;\gamma_h)\le \lambda_{2,\alpha}(B;\gamma_h)\), and equality holds if and only if \(\Omega=B\).

Choices:
B. \(\lambda_{2,\alpha}(\Omega;\gamma_h)\le \lambda_{2,\alpha}(B;\gamma_h)\) for every \(\alpha\le 0\), and equality holds if and only if \(\Omega=B\).
C. \(\lambda_{2,\alpha}(\Omega;\gamma_h)\le \lambda_{2,\alpha}(B;\gamma_h)\).
D. \(\lambda_{2,\alpha}(\Omega;\gamma_h)\le \lambda_{2,\alpha}(B;\gamma_h)\) whenever \(\alpha\in[-\sigma_1(\Omega;\gamma_h),0]\), and equality holds if and only if \(\Omega=B\).
E. \(\lambda_{2,\alpha}(\Omega;\gamma_h)\ge \lambda_{2,\alpha}(B;\gamma_h)\), and equality holds if and only if \(\Omega=B\).


\begin{Theorem}\label{ILR2n}
Let $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then, for every embedding $f$ of $K$ into ${\mathbb R}^{2n}$, there exists an element $\lambda$ in $\Lambda^{n-1,n}(K)$ such that ${\rm lk}_{2}(f(\lambda)) = 1$.
\end{Theorem}

Throughout this paper, we work in the piecewise linear category. For the fundamentals of piecewise-linear topology, we refer the reader to \cite{H69}, \cite{RS82}. Let $K$ be a finite simplicial $n$-complex, which we identify with its polyhedron in this context. For an integer $k\le n$, let $\varDelta^{k}(K)$ denote the set of all $k$-simplices in $K$. Then the subcomplex $\bigcup_{l\le k}\varDelta^{l}(K)$ of $K$ is called the {\it $k$-skeleton} of $K$ and denoted by $K^{k}$. Let $f$ be an embedding of $K$ into $\mathbb{R}^{m}$. Let $\Lambda^{p,q}(K)$ be the set of all unordered pairs $\{\gamma, \gamma'\}$ of mutually disjoint subcomplexes of $K$ such that $\gamma$ is homeomorphic to the $p$-sphere and $\gamma'$ is homeomorphic to the $q$-sphere. We identify any pair $(\gamma,\gamma')$ in $\Lambda^{p,q}(K)$ with the disjoint union $\gamma\sqcup \gamma'$. Then, for any $\lambda = \gamma \sqcup \gamma' \in \Lambda^{p,q}(K)$, the image $f(\lambda) = f(\gamma) \sqcup f(\gamma')$ forms a two-component link in $f(K)$ consisting of a $p$-sphere and a $q$-sphere. If $p+q = m-1$, the \textit{$\mathbb{Z}_2$-linking number} ${\rm lk}_2(L) = {\rm lk}_2(S^p, S^q) = {\rm lk}_2(S^q, S^p) \in \mathbb{Z}_2$ of a two-component link $L = S^p \sqcup S^q$ in ${\mathbb R}^m$ is well-defined (cf.\ \cite{ST80}). If $m=2n+1$, there are known examples of simplicial $n$-complexes with the following property: every embedding into $\mathbb{R}^{2n+1}$ necessarily contains a two-component link of $n$-spheres with $\mathbb{Z}_2$-linking number $1$. Such examples were constructed by Segal--Spie\.{z} \cite{SeSp92}, Lovasz--Schrijver \cite{LS98}, Skopenkov \cite{Sk03}, Taniyama \cite{T00}, and more recently by the author \cite{N26}. In particular, the case of $n=1$ is famous as the Conway--Gordon--Sachs theorem (\cite{CG83}, \cite{S84}), and such a simplicial $1$-complex (i.e., a graph) is said to be {\it intrinsically linked}.

Our purpose in this paper is to present an example of a minimal simplicial $n$-complex that possesses such an intrinsic linking property with respect to embeddings into $\mathbb{R}^{2n}$. Let us introduce three simplicial $n$-complexes $N_{1}^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ as follows. First, let $\sigma_{m}=|a_{0}a_{1}\cdots a_{m}|$ be an $m$-simplex whose vertices are $a_{0},a_{1},\ldots,a_{m}$. This can be naturally regarded as a simplicial $m$-complex consisting of itself and all its faces. In this setting, the $k$-skeleton of $\sigma_{m}$ is standardly denoted by $\sigma_{m}^{k}$ for $k\le m$. Then we define
\begin{eqnarray*}
N_{1}^{(n)} = \sigma_{2n+2}^{n}\setminus \{|a_{0}a_{j_{1}}\cdots a_{j_{n}}|\ |\ 1\le j_{1}<j_{2}<\cdots<j_{n}\le 2n+2\}.
\end{eqnarray*}
In other words, $N_{1}^{(n)}$ is the subcomplex of $\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices that contain the vertex $a_0$. Next, for a positive integer $k$, let $[k]^{*m}$ denote the $m$-fold join of $k$ points. Explicitly, we set
\begin{eqnarray*}
[k]^{*m} =
\{a_{0}^{0},a_{1}^{0},\ldots,a_{k-1}^{0}\} * \{a_{0}^{1},a_{1}^{1},\ldots,a_{k-1}^{1}\} * \cdots * \{a_{0}^{n},a_{1}^{n},\ldots,a_{k-1}^{n}\}.
\end{eqnarray*}
Note that $[k]^{*n+1}$ can naturally be regarded as a simplicial $n$-complex. Then we define
\begin{eqnarray*}
N_{2}^{(n)} = [3]^{*n+1}\setminus \{|a_{0}^{0}a_{j_{1}}^{1}\cdots a_{j_{n}}^{n}|\ |\ j_{1},j_{2},\ldots,j_{n}\in\{0,1,2\}\}.
\end{eqnarray*}
In other words, $N_{2}^{(n)}$ is the subcomplex of $[3]^{*n+1}$ obtained by removing all $n$-simplices that contain the vertex $a_{0}^{0}$. Finally, we define
\begin{eqnarray*}
N_{3}^{(n)} = \sigma_{2n+2}^{n}\setminus \varDelta^{n}(\sigma_{n+1}).
\end{eqnarray*}
In other words, $N_{3}^{(n)}$ is the subcomplex of $\sigma_{2n+2}^{n}$ obtained by removing all $n$-simplices contained in $\sigma_{n+1}=|a_{0}a_{1}\cdots a_{n+1}|$. As we will see in the next section, it is well known that neither $\sigma_{2n+2}^{n}$ nor $[3]^{*n+1}$ cannot be embedded in ${\mathbb R}^{2n}$ (Theorem \ref{VKF}). On the other hand, it is also known that any proper subcomplex of $\sigma_{2n+2}^{n}$ and of $[3]^{*n+1}$ can be embedded in $\mathbb{R}^{2n}$ (Gr\"{u}nbaum \cite{Grun69}). Therefore, each of $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ is embeddable in $\mathbb{R}^{2n}$. Then we have the following:

\begin{Theorem}{\rm (van Kampen \cite{VK33}, Flores \cite{Flores32})}\label{VKF}
Let $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\varphi$ of $K$ into ${\mathbb R}^{2n}$, the following holds:
\begin{eqnarray*}
\sum_{\substack{s,s'\in\varDelta^{n}(K)\\ s\cap s'=\varnothing}}l(\varphi(s),\varphi(s'))\equiv 1\pmod{2}.
\end{eqnarray*}
\end{Theorem}

Therefore, $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ are simplicial $n$-complexes that are `intrinsically linked' with respect to embeddings into ${\mathbb R}^{2n}$. We note that the simplicial $n$-complex $\sigma_{2n+2}^{n}\setminus\{|a_{0}a_{1}\cdots a_{n}|\}$ has already been shown to satisfy this property (Freedman--Krushkal \cite[Proposition 4.1]{FK14}; see also Freedman--Krushkal--Teichner \cite[Lemma 6]{FKT94}). However, both our $N_{1}^{(n)}$ and $N_{3}^{(n)}$ are proper subcomplexes of this complex. Moreover, the complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ are minimal in the following sense among all simplicial $n$-complexes possessing this property.

\begin{Proposition}\label{min}
Let $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\to {\mathbb R}^{2n}$ such that ${\rm lk}_{2}(f(\lambda))= 0$ for any element $\lambda$ in $\Lambda^{n-1,n}(N)$.
\end{Proposition}

\begin{Remark}\label{nsp}
A graph $G$ is said to be {\it non-separating planar} if there exists an embedding $f: G \to \mathbb{R}^2$ such that, for every cycle $\gamma$ in $G$, all vertices of $G$ not lying on $\gamma$ belong to the same connected component of $\mathbb{R}^2 \setminus f(\gamma)$. Equivalently, $G$ is not non-separating planar if and only if for every embedding $f:G\to \mathbb{R}^2$, the image $f(G)$ contains a two-component link consisting of a $0$-sphere and a $1$-sphere with $\mathbb{Z}_2$-linking number $1$. Dehkordi--Farr proved in \cite{DF} that a graph is non-separating planar if and only if it contains no subgraph homeomorphic to $K_1 \sqcup K_4$, $K_1 \sqcup K_{2,3}$, or $K_{1,1,3}$ as illustrated in Fig. \ref{nonsp}. They further applied this characterization to the study of linkless embeddings of graphs in $\mathbb{R}^3$. For related results, see Fowler--Li--Pavelescu \cite{FLP23}, Pavelescu--Pavelescu \cite{PP24}. Note that $N_1^{(1)}$ is isomorphic to $K_1 \sqcup K_4$, $N_2^{(1)}$ is isomorphic to $K_1 \sqcup K_{2,3}$, and $N_3^{(1)}$ is isomorphic to $K_{1,1,3}$. Thus, our simplicial $n$-complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ may be regarded as higher-dimensional analogues of the minimal subgraphs that prevent a graph from being non-separating planar.
\end{Remark}

\begin{Theorem}{\rm (van Kampen \cite{VK33}, Flores \cite{Flores32})}\label{VKF}
Let $n$ be a positive integer, and let $K$ be a simplicial $n$-complex that is $\sigma_{2n+2}^{n}$ or $[3]^{*n+1}$. Then for every generic immersion $\varphi$ of $K$ into ${\mathbb R}^{2n}$, the following holds:
\begin{eqnarray*}
\sum_{\substack{s,s'\in\varDelta^{n}(K)\\ s\cap s'=\varnothing}}l(\varphi(s),\varphi(s'))\equiv 1\pmod{2}.
\end{eqnarray*}
\end{Theorem}

\begin{proof}[Proof of Theorem \ref{ILR2n}]
We first consider the case $K = N_1^{(n)}$. In what follows, for an $m$-simplex $s$, we will also denote its $(m-1)$-skeleton $s^{m-1}$ by $\partial s$. We define collections $\Gamma^{n-1}$ and $\Gamma^n$ of $(n-1)$ and $n$-subcomplexes of $N_1^{(n)}$, respectively, as follows:
\begin{eqnarray*}
\Gamma^{n-1} &=& \left\{\partial s \mid s\in \varDelta^{n}(\sigma_{2n+2}^{n}),\,a_{0}\in s\right\},\\
\Gamma^{n} &=& \left\{\partial t \mid t\in \varDelta^{n+1}(\sigma_{2n+2}),\,a_{0}\notin t\right\}.
\end{eqnarray*}
Then every $\lambda \in \Lambda^{n-1,n}(N_1^{(n)})$ is the disjoint union of an element $\gamma$ of $\Gamma^{n-1}$ and an element $\gamma'$ of $\Gamma^n$, see Fig. \ref{N_1_link} for $n=2$. Let $f:N_{1}^{(n)}\to {\mathbb R}^{2n}$ be an embedding. This can be extended to a generic immersion $\hat{f}:\sigma_{2n+2}^{n} \to \mathbb{R}^{2n}$ such that, for any two $n$-simplices $s,s' \in \varDelta^n(\sigma_{2n+2}^{n})$ with $a_0 \in s$ and $a_0 \notin s'$, the images $\hat{f}(s)$ and $\hat{f}(s')$ intersect transversely at finitely many double points, all of which occur in the interiors of both $n$-simplices. Note that for any $\lambda = \gamma \sqcup \gamma' \in \Lambda^{n-1,n}(N_1^{(n)})$ with $\gamma = \partial s$, $s \in \varDelta^n(\sigma_{2n+2}^n)$ and $a_0 \in s$, we have
\begin{eqnarray}\label{lkdef}
{\rm lk}_{2}(f(\gamma),f(\gamma'))
\equiv \sum_{s'\in \varDelta^{n}(\gamma')}l(\hat{f}(s),\hat{f}(s'))\pmod{2}.
\end{eqnarray}
On the other hand, for any two disjoint $n$-simplices $s, s' \in \varDelta^n(\sigma_{2n+2}^n)$ with $a_0 \in s$ and $a_0 \notin s'$, there uniquely exists an element $\lambda = \gamma \sqcup \gamma' \in \Lambda^{n-1,n}(N_1^{(n)})$ such that $\gamma = \partial s$ and $\gamma' \supset s'$. Then, by (\ref{lkdef}) and Theorem \ref{VKF}, we obtain
\begin{eqnarray*}\label{lksum}
\sum_{\lambda\in \Lambda^{n-1,n}(N_{1}^{(n)})}{\rm lk}_{2}(f(\lambda))
&=& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{1}^{(n)}) \\ \gamma\in \Gamma^{n-1},\,\gamma'\in\Gamma^{n}}}{\rm lk}_{2}(f(\gamma),f(\gamma'))\\
&\equiv& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{1}^{(n)}) \\ \gamma=\partial s,\,s\in \varDelta^{n}(\sigma_{2n+2}^{n}),\,a_{0}\in s}}
\bigg(\sum_{s'\in \varDelta^{n}(\gamma')}l(\hat{f}(s),\hat{f}(s'))\bigg)\nonumber\\
&=& \sum_{\substack{s,\,s'\in \varDelta^{n}(\sigma_{2n+2}^{n}) \\ s\cap s'=\varnothing}}
l(\hat{f}(s),\hat{f}(s'))\nonumber\\
&\equiv& 1\pmod{2}. \nonumber
\end{eqnarray*}
Thus, there exists some $\lambda$ in $\Lambda^{n-1,n}(N_{1}^{(n)})$ such that ${\rm lk}_{2}(f(\lambda))=1$.

Next, we consider the case $K = N_{2}^{(n)}$. We define collections $\Gamma^{n-1}$ and $\Gamma^n$ of $(n-1)$ and $n$-subcomplexes of $N_{2}^{(n)}$, respectively, as follows:
\begin{eqnarray*}
\Gamma^{n-1} &=& \left\{\partial s \mid s\in \varDelta^{n}([3]^{*n+1}),\,a_{0}^{0}\in s\right\},\\
\Gamma^{n} &=& \left\{\left\{a_{1}^{0},a_{2}^{0}\right\}*\left\{a_{i_{1}}^{1},a_{j_{1}}^{1}\right\}*\cdots*\left\{a_{i_{n}}^{n},a_{j_{n}}^{n}\right\} \mid 0\le i_{q}<j_{q}\le 2,\,q\ge 1\right\}.
\end{eqnarray*}
Note that the $(n+1)$-fold join of two points is homeomorphic to the $n$-sphere. Then every $\lambda \in \Lambda^{n-1,n}(N_{2}^{(n)})$ is the disjoint union of an element $\gamma$ of $\Gamma^{n-1}$ and an element $\gamma'$ of $\Gamma^n$, see Fig. \ref{N_2_link} for $n=2$. Let $f:N_{2}^{(n)}\to {\mathbb R}^{2n}$ be an embedding. This can be extended to a generic immersion $\hat{f}:[3]^{*n+1} \to \mathbb{R}^{2n}$ such that, for any two $n$-simplices $s,s' \in \varDelta^n([3]^{*n+1})$ with $a_{0}^{0} \in s$ and $a_{0}^{0} \notin s'$, the images $\hat{f}(s)$ and $\hat{f}(s')$ intersect transversely at finitely many double points, all of which occur in the interiors of both $n$-simplices. Then, in a similar way as in the case of $N_{1}^{(n)}$, we have
\begin{eqnarray*}\label{lksum2}
\sum_{\lambda\in \Lambda^{n-1,n}(N_{2}^{(n)})}{\rm lk}_{2}(f(\lambda))
&=& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{2}^{(n)}) \\ \gamma\in \Gamma^{n-1},\,\gamma'\in\Gamma^{n}}}{\rm lk}_{2}(f(\gamma),f(\gamma'))\\
&\equiv& \sum_{\substack{\gamma\sqcup\gamma'\in \Lambda^{n-1,n}(N_{2}^{(n)}) \\ \gamma=\partial s,\,s\in \varDelta^{n}([3]^{*n+1}),\,a_{0}^{0}\in s}}
\bigg(\sum_{s'\in \varDelta^{n}(\gamma')}l(\hat{f}(s),\hat{f}(s'))\bigg)\nonumber\\
&=& \sum_{\substack{s,\,s'\in \varDelta^{n}([3]^{*n+1}) \\ s\cap s'=\varnothing}}
l(\hat{f}(s),\hat{f}(s'))\nonumber\\
&\equiv& 1\pmod{2}. \nonumber
\end{eqnarray*}
Thus the result follows.

Therefore, $N_1^{(n)}$, $N_2^{(n)}$ and $N_3^{(n)}$ are simplicial $n$-complexes that are `intrinsically linked' with respect to embeddings into ${\mathbb R}^{2n}$. We note that the simplicial $n$-complex $\sigma_{2n+2}^{n}\setminus\{|a_{0}a_{1}\cdots a_{n}|\}$ has already been shown to satisfy this property (Freedman--Krushkal \cite[Proposition 4.1]{FK14}; see also Freedman--Krushkal--Teichner \cite[Lemma 6]{FKT94}). However, both our $N_{1}^{(n)}$ and $N_{3}^{(n)}$ are proper subcomplexes of this complex. Moreover, the complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ are minimal in the following sense among all simplicial $n$-complexes possessing this property.

\begin{Proposition}\label{min}
Let $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\to {\mathbb R}^{2n}$ such that ${\rm lk}_{2}(f(\lambda))= 0$ for any element $\lambda$ in $\Lambda^{n-1,n}(N)$.
\end{Proposition}

A central role in the proof of Theorem \ref{ILR2n} is played by the van Kampen--Flores theorem. In the next section, we recall this theorem and present proofs of Theorem \ref{ILR2n} and Proposition \ref{min} using it. The van Kampen--Flores theorem is also used to construct examples of simplicial $n$-complexes that are intrinsically linked with respect to embeddings into $\mathbb{R}^{2n+1}$, see Nikkuni \cite{N26}.

\begin{Remark}\label{nsp}
A graph $G$ is said to be {\it non-separating planar} if there exists an embedding $f: G \to \mathbb{R}^2$ such that, for every cycle $\gamma$ in $G$, all vertices of $G$ not lying on $\gamma$ belong to the same connected component of $\mathbb{R}^2 \setminus f(\gamma)$. Equivalently, $G$ is not non-separating planar if and only if for every embedding $f:G\to \mathbb{R}^2$, the image $f(G)$ contains a two-component link consisting of a $0$-sphere and a $1$-sphere with $\mathbb{Z}_2$-linking number $1$. Dehkordi--Farr proved in \cite{DF} that a graph is non-separating planar if and only if it contains no subgraph homeomorphic to $K_1 \sqcup K_4$, $K_1 \sqcup K_{2,3}$, or $K_{1,1,3}$ as illustrated in Fig. \ref{nonsp}. They further applied this characterization to the study of linkless embeddings of graphs in $\mathbb{R}^3$. For related results, see Fowler--Li--Pavelescu \cite{FLP23}, Pavelescu--Pavelescu \cite{PP24}. Note that $N_1^{(1)}$ is isomorphic to $K_1 \sqcup K_4$, $N_2^{(1)}$ is isomorphic to $K_1 \sqcup K_{2,3}$, and $N_3^{(1)}$ is isomorphic to $K_{1,1,3}$. Thus, our simplicial $n$-complexes $N_{1}^{(n)}$, $N_{2}^{(n)}$ and $N_{3}^{(n)}$ may be regarded as higher-dimensional analogues of the minimal subgraphs that prevent a graph from being non-separating planar.
\end{Remark}

\begin{figure}[htbp]
\begin{center}
\scalebox{0.525}{\includegraphics*{nonsp}}
\caption{$K_1 \sqcup K_4$, $K_1 \sqcup K_{2,3}$ and $K_{1,1,3}$}
\label{nonsp}
\end{center}
\end{figure}

A proof sketch is not given here beyond indicating the main tool: “A central role in the proof of Theorem \ref{ILR2n} is played by the van Kampen--Flores theorem.” The authors say that “In the next section, we recall this theorem and present proofs of Theorem \ref{ILR2n} and Proposition \ref{min} using it.”

A proof sketch is not given here beyond indicating the main tool: “A central role in the proof of the main theorem is played by the van Kampen--Flores theorem.” The authors say that “Next we recall this theorem and present proofs of the main theorem and \begin{Proposition}\label{min}
Let $N$ be a proper subcomplex of $N_1^{(n)}$, $N_2^{(n)}$ or $N_3^{(n)}$. Then there exists an embedding $f:N\to {\mathbb R}^{2n}$ such that ${\rm lk}_{2}(f(\lambda))= 0$ for any element $\lambda$ in $\Lambda^{n-1,n}(N)$.
\end{Proposition} using it.”

Question:
Let \(n\) be a positive integer. Write \(\sigma_m=|a_0a_1\cdots a_m|\) for the standard \(m\)-simplex, and let \(\sigma_m^n\) denote its \(n\)-skeleton. Define
\[
N_1^{(n)}=\sigma_{2n+2}^n\setminus\{\text{all }n\text{-simplices containing }a_0\},
\]
and, with
\[
[3]^{*(n+1)}=\{a_0^0,a_1^0,a_2^0\}*\{a_0^1,a_1^1,a_2^1\}*\cdots*\{a_0^n,a_1^n,a_2^n\},
\]
define
\[
N_2^{(n)}=[3]^{*(n+1)}\setminus\{\text{all }n\text{-simplices containing }a_0^0\},
\]
and
\[
N_3^{(n)}=\sigma_{2n+2}^n\setminus \Delta^n(\sigma_{n+1}),\quad \sigma_{n+1}=|a_0a_1\cdots a_{n+1}|,
\]
so \(N_3^{(n)}\) is obtained from \(\sigma_{2n+2}^n\) by removing all \(n\)-simplices contained in \(\sigma_{n+1}\). For a simplicial complex \(K\), let \(\Lambda^{p,q}(K)\) be the set of unordered pairs \(\lambda=\gamma\sqcup\gamma'\) of mutually disjoint subcomplexes of \(K\) such that \(\gamma\cong S^p\) and \(\gamma'\cong S^q\). If \(p+q=m-1\), let \({\rm lk}_2\) denote the \(\mathbb Z_2\)-linking number of a two-component link \(S^p\sqcup S^q\subset \mathbb R^m\). Suppose \(K\) is one of \(N_1^{(n)},N_2^{(n)},N_3^{(n)}\). Which statement holds for every embedding \(f:K\to \mathbb R^{2n}\)?

There exists an element \(\lambda\in \Lambda^{n-1,n}(K)\) such that \({\rm lk}_2(f(\lambda))=1\); equivalently, every embedding \(f:K\to\mathbb R^{2n}\) contains a pair of disjoint subcomplexes \(\gamma\cong S^{n-1}\) and \(\gamma'\cong S^n\) whose images form a two-component link in \(\mathbb R^{2n}\) with nontrivial \(\mathbb Z_2\)-linking number.

Choices:
B. There exists an element \(\lambda\in \Lambda^{n,n}(K)\) such that \({\rm lk}_2(f(\lambda))=1\); equivalently, every embedding \(f:K\to\mathbb R^{2n}\) contains a pair of disjoint subcomplexes \(\gamma\cong S^{n}\) and \(\gamma'\cong S^n\) whose images form a two-component link in \(\mathbb R^{2n}\) with nontrivial \(\mathbb Z_2\)-linking number.
C. There exists an element \(\lambda\in \Lambda^{n-1,n}(K)\) such that the images of its two components under \(f\) form a non-split two-component link in \(\mathbb R^{2n}\).
D. There exists a single element \(\lambda\in \Lambda^{n-1,n}(K)\), depending only on \(K\) and not on the embedding \(f\), such that \({\rm lk}_2(f(\lambda))=1\) for every embedding \(f:K\to\mathbb R^{2n}\).
E. For every embedding \(f:K\to\mathbb R^{2n}\) and every element \(\lambda\in \Lambda^{n-1,n}(K)\), one has \({\rm lk}_2(f(\lambda))=1\); equivalently, every pair of disjoint subcomplexes \(\gamma\cong S^{n-1}\) and \(\gamma'\cong S^n\) in \(K\) is sent to a two-component link with nontrivial \(\mathbb Z_2\)-linking number.

sphere-dimension pair must be \((n-1,n)\) in \(\mathbb R^{2n}\), not \((n,n)\)

dropped the explicit conclusion \({\rm lk}_2=1\), keeping only existence of a non-split \((S^{n-1},S^n)\)-link

dependence of the witnessing \(\lambda\) on the embedding \(f\)

parity argument yields existence of at least one linked pair, not all pairs


\begin{theorem}\label{geometricallyFiniteNearOrbit}
    Suppose that a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if there exists an escaping sequence of hyperbolic elements in $G$.
\end{theorem}

\begin{lemma}\label{producingHyperbolicElements}
    Let the action of $G$ on $M$ be a non-elementary convergence group action.
    If there exist an element $f \in G$ and an open set $U \subset M$ such that $f\overline{U} \subsetneqq U$, then $f$ is hyperbolic.
\end{lemma}

Let $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\partial X$, which decomposes $\partial X$ into the \textit{limit set} $\Lambda G$ and the domain of discontinuity. Let $\Pi \subset \Lambda G$ be the set of parabolic points. The action of $G$ on $X$ is called \textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\{B_p:p\in \Pi\}$ so that the action on the complement $\mathrm{CH}(\Lambda G)\setminus \bigcup_{p\in \Pi} B_p$ is cocompact, where $CH(\Lambda G)$ denotes the weak convex hull of $\Lambda G$ in $X$. In this case, $\Lambda G\setminus \Pi$ consists precisely of \textit{conical points}. We refer to Section \ref{preliminary} for precise definitions.

Historically, the notion of geometric finiteness originated in the theory of Kleinian groups (see \cite{BeaMas,Marden,thurstonnotes}) and has since been extended to the much broader class of groups considered here in a series of works \cite{Bow3,Bow4,bowditch2012relatively}. Since the formal introduction in \cite{Gromov}, this class of groups has been intensively studied in geometric group theory under the name of \textit{relatively hyperbolic groups}, encompassing many examples of geometric and algebraic origin. Several equivalent characterizations of relative hyperbolicity are then proposed in the literature, generalizing and going well beyond those known for Kleinian groups; see, for example, \cite{Farb, bowditch2012relatively, Osin, DruSapir, Ge1}, as well as \cite{Hru} for an overview and references therein.

The aim of this paper is to supplement these descriptions with two new characterizations in the present general setting.
Let $\{g_n\}_{n\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\{g_n\}_{n\ge 1}$ is \textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.

\label{preliminary}
In this section we introduce the preliminaries we shall use on convergence group actions (cf. \cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \cite

Let $X$ be a proper Gromov hyperbolic space, on which a countable group $G$ acts properly by isometries. The action naturally extends to a convergence group action on the Gromov boundary $\partial X$, which decomposes $\partial X$ into the \textit{limit set} $\Lambda G$ and the domain of discontinuity. Let $\Pi \subset \Lambda G$ be the set of parabolic points. The action of $G$ on $X$ is called \textit{geometrically finite} if there exists a sufficiently separated $G$-invariant family of horoballs $\{B_p:p\in \Pi\}$ so that the action on the complement $\mathrm{CH}(\Lambda G)\setminus \bigcup_{p\in \Pi} B_p$ is cocompact, where $CH(\Lambda G)$ denotes the weak convex hull of $\Lambda G$ in $X$. In this case, $\Lambda G\setminus \Pi$ consists precisely of \textit{conical points}. We refer to Section \ref{preliminary} for precise definitions.

The aim of this paper is to supplement these descriptions with two new characterizations in the present general setting.
Let $\{g_n\}_{n\ge 1}$ be a sequence of hyperbolic elements in $G$. We say that $\{g_n\}_{n\ge 1}$ is \textit{escaping} if for any point $o$ and any $R>0$, the $R$-neighborhood of $Go$ intersects the axes of only finitely many $g_n$.

The corresponding result for Kleinian groups was proved by Bonahon \cite{bonahon-bouts} and was recently generalized to pinched negatively curved manifolds by Kapovich and Liu \cite{KL19}. In contrast to these works, our approach relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma.

\begin{theorem}\label{geometricallyFiniteNonconicalCountable}
 Suppose a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if the set of non-conical points is uncountable.
\end{theorem}

Finally, let us conclude the Introduction with an open question.
By a theorem of Yaman \cite{Yaman} (and of Bowditch \cite{Bow98} in the cocompact case), geometrically finite actions on Gromov hyperbolic spaces can be recovered from the dynamics on the boundary. More precisely, if a convergence group action is geometrically finite (i.e., its limit set consists only of conical points and bounded parabolic points) then the group admits a geometrically finite action on some proper Gromov hyperbolic space. Moreover, the induced action on the boundary coincides with the original convergence group action. The following question asks whether the assumption of bounded parabolic points could be weakend as follows.

\begin{question}\label{geometricallyFiniteConvergenceGroup}
 Suppose a discrete group $G$ admits a convergence group action on a compact metrizable space with countably many non-conical limit points. Does $G$ admit a geometrically finite action on some Gromov hyperbolic space?
\end{question}

Together with the standing assumption that the action is non-elementary (which ensures the existence of independent hyperbolic elements), we have a finiteness statement about conjugacy classes.
\begin{lemma}\label{stableTranslationLength}
 For any $r > 0$, there are only finitely many conjugacy classes of hyperbolic elements in $G$ with translation length less than $r$.
\end{lemma}
\begin{proof}
 Let $r > 0$, and let $g \in G$ be a hyperbolic element with $\tau(g) < r$. By our assumption, there exists $h \in G$ and an axis $\gamma$ of $g$ such that $d(h o, \gamma) \le R_0$. Thus, we can choose a point $x \in \gamma$ satisfying $d(h o, x) \le R_0$. Applying the triangle inequality and using Lemma~\ref{translationDistance}, we estimate:
 $$
 d(h o, g h o) \le d(h o, x) + d(x, g x) + d(g x, g h o) \le R_0 + \tau(g) + D_1 + R_0 < 2R_0 + D_1 + r.
 $$
 This yields $d(o, h^{-1} g h \, o) = d(h o, g h o) < 2R_0 + D_1 + r$.
 Since the action of $G$ on $X$ is proper, the set $\{ k \in G : d(o, k o) < 2R_0 + D_1 + r \}$ is finite. Therefore, the set of conjugates $h^{-1}gh$ belongs to this finite set. Consequently, there are only finitely many distinct conjugacy classes of such elements $g$.
\end{proof}

Throughout this section, let $G$ be a discrete group that admits a non-elementary, geometrically infinite, proper action on a proper $\delta$-hyperbolic space $X$ for some $\delta\ge 0$, and fix a basepoint $o\in X$.
By Lemma~\ref{localQuasiGeodesic}, there exist constants $L>16\delta$, $\lambda \ge 1$, and $C\ge 0$ such that every $L$-local $(3, 2D_0)$-quasi-geodesic in $X$ is a $(\lambda, C)$-quasi-geodesic.
By Lemma~\ref{MorseLemma}, there exists a constant $B>0$ such that $d_{\text{Haus}}(p, [p_-, p_+])\le B$ for every $(\lambda, C)$-quasi-geodesic $p$.

\label{preliminary}
In this section we introduce the preliminaries we shall use on convergence group actions (cf. \cite{tukia1994convergence, bowditch1999convergence}) and hyperbolic spaces (cf. \cite

The corresponding result for Kleinian groups was proved by Bonahon \cite{bonahon-bouts} and was recently generalized to pinched negatively curved manifolds by Kapovich and Liu \cite{KL19}. In contrast to these works, our approach relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma.

Following the strategy of Bishop \cite{Bishop96} for Kleinian groups, we derive the second characterization of geometric infiniteness  from Theorem \ref{geometricallyFiniteNearOrbit}. This refines the result of Beardon and Maskit \cite{BeaMas} in terms of non-conical limit sets.

\begin{theorem}\label{geometricallyFiniteNonconicalCountable}
    Suppose  a group $G$ admits a non-elementary proper action on a proper Gromov hyperbolic space $X$. Then the action is geometrically infinite if and only if the set of non-conical points is uncountable.
\end{theorem}

We remark that  the negatively pinched  manifold case was due to  Kapovich and Liu   \cite{KL20} after their work \cite{KL19} on Theorem \ref{geometricallyFiniteNearOrbit}. To put into perspective, for finitely generated geometrically infinite Kleinian groups, the non‑conical limit set has   maximal Hausdorff dimension whose precise value was computed by Bishop–Jones \cite{BJ97bddgeo}, subsequent contributions \cite{KL20}, and recently completed in \cite{mj2025hausdorff} in full general case. The question of whether Theorem~\ref{geometricallyFiniteNonconicalCountable} holds in this level of generality arose naturally in the course of that study.

Finally, let us conclude the Introduction with an open question.
By a theorem of Yaman \cite{Yaman} (and of Bowditch \cite{Bow98} in the cocompact case), geometrically finite actions on Gromov hyperbolic spaces can be recovered from the dynamics on the boundary. More precisely, if a convergence group action is geometrically finite (i.e., its limit set consists only of conical points and bounded parabolic points) then the group admits a geometrically finite action on some proper Gromov hyperbolic space. Moreover, the induced action on the boundary coincides with the original convergence group action. The following question asks whether the assumption of bounded parabolic points could be weakend as follows.

\begin{question}\label{geometricallyFiniteConvergenceGroup}
    Suppose a  discrete group  $G$ admits a convergence group action on a compact metrizable space with  countably many non-conical limit points. Does $G$ admit a geometrically finite action on some Gromov hyperbolic space?
\end{question}

\begin{remark}

\begin{enumerate}
\item
This is related to a question of Kapovich \cite[Problem 7]{Kap05} asking whether every convergence group action can be realized as the boundary action of a proper action on some Gromov hyperbolic space. If so, then a positive answer would follow from Theorem~\ref{geometricallyFiniteNonconicalCountable} together with Yaman’s theorem.
Moreover, in \cite{sun2019dynamicalcharacterizationacylindricallyhyperbolic}, Bin Sun proved that the action constructed in \cite{Bow98} on a hyperbolic space contains WPD (i.e. weak proper discontinuity) elements.

\item The assumption that the action is a convergence group action is necessary. It is easy to give examples of \emph{non-relatively hyperbolic} groups $G$ which admit a minimal action on a compact metriziable space $M$ so that all but {countably many points} $\xi\in M$ are \emph{conical points} in the usual sense.
\begin{itemize}
    \item  there exist a sequence $\{g_n\}$ in $G$ and distinct points $a, b \in M$ such that $g_n x \to b$ locally uniformly for all $x \in M \setminus \{\xi\}$, while $g_n \xi \to a$.
\end{itemize}
Indeed, any non-elementary action on a locally infinite tree $T$ gives raise to the desired action on the compact space $M=\partial T\cup T^0$, the union of the ends of the tree and the vertex set. We equip $M$ with Bowditch topology  (since $T$ is a fine graph in sense of \cite{bowditch2012relatively}). It is easy to check that the ends of the tree are conical points in the above sense.
\end{enumerate}

\end{remark}
\subsection*{Structure of the paper}
Section 2 reviews the necessary preliminaries on convergence group actions and $\delta$-hyperbolic spaces. A key technical lemma (Lemma~\ref{producingHyperbolicElements}) for producing hyperbolic elements under a dynamical condition is presented.
In Section 3, assuming the condition in Theorem~\ref{geometricallyFiniteNearOrbit}, we construct geometric objects analogous to those in a cusp-uniform action. We prove that every point in the limit set $\Lambda G$ is either conical or bounded parabolic, thereby establishing the theorem.
In Section 4, we largely follow the strategy of \cite{Bishop96} to construct uncountably many non-conical limit points under the hypothesis of geometric infiniteness. The proof uses the tool of $L$-local quasi-geodesics to build the required paths, thus proving Theorem~\ref{geometricallyFiniteNonconicalCountable}.

\subsection*{Acknowledgments}
The second-named author is grateful to Mahan MJ  for insightful discussions about  Theorem \ref{geometricallyFiniteNonconicalCountable}.

A proof sketch for Theorem~\ref{geometricallyFiniteNearOrbit} is indicated in the “Structure of the paper” paragraph: in Section~3, “assuming the condition in Theorem~\ref{geometricallyFiniteNearOrbit}, we construct geometric objects analogous to those in a cusp-uniform action” and then “prove that every point in the limit set $\Lambda G$ is either conical or bounded parabolic, thereby establishing the theorem.” The introduction also notes that the approach “relies only on elementary arguments in hyperbolic geometry and does not invoke the Margulis lemma,” and that a “key technical lemma (Lemma~\ref{producingHyperbolicElements}) for producing hyperbolic elements under a dynamical condition is presented” in Section~2.

Question: Let a group $G$ act non-elementarily and properly by isometries on a proper Gromov hyperbolic space $X$. Write $\Lambda G\subset \partial X$ for the limit set, $\Pi\subset \Lambda G$ for the parabolic points, and $CH(\Lambda G)$ for the weak convex hull of $\Lambda G$. The action is called geometrically finite if there exists a sufficiently separated $G$-invariant family of horoballs $\{B_p:p\in \Pi\}$ such that the action on $CH(\Lambda G)\setminus \bigcup_{p\in \Pi} B_p$ is cocompact; geometrically infinite means not geometrically finite. A sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements of $G$ is called escaping if, for every $o\in X$ and every $R>0$, the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$. Which of the following statements is equivalent to the action being geometrically infinite?

There exists an escaping sequence of hyperbolic elements in $G$.

Choices:
B. For every sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements in $G$ whose conjugacy classes are pairwise distinct, the sequence is escaping.
C. There exists a sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements in $G$ such that, for some $o\in X$ and some $R>0$, the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$.
D. There exists a sequence $\{g_n\}_{n\ge 1}$ of hyperbolic elements in $G$ such that, for every $o\in X$, there exists $R>0$ for which the $R$-neighborhood of the orbit $Go$ intersects the axes of only finitely many $g_n$.
E. There exists an escaping sequence $\{g_n\}_{n\ge 1}$ of loxodromic isometries of $X$, not necessarily contained in $G$.

existential escaping sequence replaced by universal claim over distinct conjugacy classes

dropped the quantifier requirement 'for every $o$ and every $R>0$'

quantifier order on $R$ weakened from 'for every $R>0$' to 'there exists $R>0$'

sequence no longer required to lie in the acting group $G$


The evolution of hypersurfaces by their mean curvature has been studied extensively since Huisken \cite{huisken84}, who proved that closed embedded convex hypersurfaces in Euclidean space contract to round points under mean curvature flow.
More recently, the mean curvature flow of submanifolds with higher codimension has been studied. The first author and Baker \cite{andrewsbaker2010} proved an analogous result to that of \cite{huisken84} in any codimension of Euclidean background under a quadratic curvature pinching condition $|h|^2 \leq c |H|^2$ on the initial data, where $h$ is the second fundamental form, $H$ is the mean curvature, and $c$ is a dimensional constant. Since then, Baker and Nguyen \cite{bakernguyen2023} have explored the flow in the case where the background is a sphere of arbitrary codimension, where the sharp quartic pinching condition for the sphere is found in \cite{vogiatzi2024sharpquarticpinchingmean}. Other directions in high codimension have been explored, including ancient solutions \cite{lynchnguyen2021, Risa_2018}, singularity formation \cite{vogiatzi2023singularitymodelshighcodimension}, flow in complex projective space \cite{vogiatzi2023meancurvatureflowhigh}, and flow in pseudo-Euclidean space of submanifolds with maximal spacelike dimension \cite{Lambert_2019}.

In this paper, we consider the mean curvature flow (MCF) of embedded spacelike submanifolds under the assumption that the normal subspace at each point has signature $(1,k)$, so that the codimension may be arbitrary but no two spacelike normal vectors are orthogonal. This implies that the cone of strictly spacelike normal vectors at any chosen point separates into two connected components. In addition, we impose the condition of \emph{spacelike convexity}, which is the condition that the acceleration vector along any geodesic of the submanifold is strictly spacelike, or equivalently that the second fundamental form $h(v,v)$ is strictly spacelike for any non-zero tangent vector $v$. At each point, the values of $h(v,v)$ must lie in the same connected component of the spacelike normal cone, which we call the \emph{inward normal cone}.
 Spacelike-convex submanifolds are a natural analogue of convex hypersurfaces in Euclidean space: For example, the orthogonal projection of such a submanifold onto any maximal spacelike subspace $\R^{n+1,0} \subset \R^{n+1,k}$ is locally convex, and the sectional curvature of the induced metric is positive.
 More interestingly for our purposes, the condition of spacelike convexity leads to a natural pinching condition for the second fundamental form: There exists $\a > 0$ such that $h(v,v) - \a H$ is inward spacelike for all unit $v\in T\S$, and similarly there exists $\b > 0$ such that $h(v,v) - \b H$ lies in the outward normal cone. We call the largest $\a$ and smallest $\b$ for which this holds the inward and outward pinching ratios respectively. Our first result is that such pinching is preserved under the flow.

\begin{proposition}
 Suppose that $\S$ is compact and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\a$ is non-decreasing in $t$, and the outward pinching ratio $\b$ is non-increasing in $t$.
\end{proposition}

In \cite{huisken84}, pinching of the form $h(v,v)\geq \alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate: For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point. In \cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior. This argument enables a very direct proof of the result of \cite{huisken84}. In this paper in Section \ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point). We say that an acausal spacelike (mean) convex embedding $F: \S^n \to \R^{n+1,k}$ is (interior) noncollapsed with parameter $\delta$ if, for any $x\in \S$, the pseudoball $\left\{|p-F(x)-\delta\frac{H(x)}{|H(x)|^2}|< \frac{\delta}{|H(x)|}\right\}$ has $F(\Sigma)$ in its exterior, touching at $F(x)$. There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\cref{noncollapsing section}).

\begin{proposition}
 Suppose that $\S$ is compact and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF with $F(.,0)$ spacelike and spacelike-convex. Then the inward pinching ratio $\a$ is non-decreasing in $t$, and the outward pinching ratio $\b$ is non-increasing in $t$.
\end{proposition}

In \cite{huisken84}, pinching of the form $h(v,v)\geq \alpha Hg(v,v)$ is preserved under the flow and can be improved via Stampaccia iteration, eventually leading to Huisken's theorem. Such an approach may be possible in our setting, but we instead use an alternative approach using non-collapsing estimates, which has the advantage of giving direct control on the global geometry of the evolving submanifolds. Sheng and Wang \cite{shengwang} proved that embedded mean-convex solutions of mean curvature flow in Euclidean space satisfy a non-collapsing estimate: For any point on the hypersurface there is a touching enclosed ball with curvature comparable to the mean curvature at that point. In \cite{andrews2011noncollapsingmeanconvexmeancurvature}, the first author found a proof of such an estimate using a direct maximum principle argument, which also provides a corresponding result for balls touching from the exterior. This argument enables a very direct proof of the result of \cite{huisken84}. In this paper in Section \ref{noncollapsing section} we prove a version of noncollapsing for space-like convex submanifolds (the first high-codimension setting where such a result is known). This non-collapsing result holds also for the more general class of acausal submanifolds which are spacelike-mean-convex (that is, the mean curvature is strictly spacelike at each point). We say that an acausal spacelike (mean) convex embedding $F: \S^n \to \R^{n+1,k}$ is (interior) noncollapsed with parameter $\delta$ if, for any $x\in \S$, the pseudoball $\left\{|p-F(x)-\delta\frac{H(x)}{|H(x)|^2}|< \frac{\delta}{|H(x)|}\right\}$ has $F(\Sigma)$ in its exterior, touching at $F(x)$. There is a similar definition of exterior noncollapsing. Our second result is that the flow preserves noncollapsing (\cref{noncollapsing section}).

The assumption of turning number $1$ is carried out throughout the paper. The Euclidean inner product leads to a notion of long time existence, hence a subsequence convergence result that a suitably rescaled flow admits a subsequence that converges smoothly to a sphere in some maximal spacelike subspace $\R^{n+1,0} \subset \R^{n+1,k}$. The approach to full convergence is motivated by first author's approach on Gauss curvature flow with the idea of considering normalised flow invariant under affine transformations \cite{andrews96} and showing that the original flow is stable near the final shape \cite[Chapter 16]{ben2020}. Since the rescaled flow consists of a Lorentz transform at each subsequent time, we consider the normalised flow that fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume. This leads to a stability argument which proves our final result: an analogue of Huisken's theorem.

\begin{proposition}[spacelike-convexity is preserved]\label{spacelike-convexity is preserved}
 Suppose that $\S$ is compact, spacelike-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then spacelike-convexity is preserved along the flow.
\end{proposition}

\begin{proposition}[Pinching is preserved]\label{pinching preserved prop}
 Suppose that $\S$ is compact, spacelike-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then there exists some $\a > 0$ such that $h(v,v) - \a H$ is spacelike for all unit $v\in T\S$ at all times.
\end{proposition}

First, we show that spacelike-mean-convexity is preserved, assuming that the induced metric stays spacelike, this condition is implied by noncollapsing so we omit the assumption for results in this section.
\begin{proposition}
 Suppose that $\S$ is compact, spacelike-mean-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then spacelike-mean-convexity is preserved along the flow.
\end{proposition}

\begin{theorem}\label{acausal spacelike-mean-convex submanifold preserves noncollapsing}
 Suppose $F: \S^n \times [0, T) \to \R^{n+1,k}$ is a spacelike-convex, compact and embedded solution to MCF, if $\S\times \{0\}$ is $\d_-$-exterior noncollapsed and $\d_+$-interior noncollapsed, then
 \begin{equation*}
 \overline{Q}(x, \d_-, t) \leq 0, \qquad \underline{Q}(x,\d_+, t) \geq 0
 \end{equation*}
 for all $t\in [0, T)$.
\end{theorem}

\begin{corollary}\label{spacelike-mean-convex data stays spacelike coro}
 Suppose that $\S$ is compact, spacelike-mean-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then $\S_t$ stays spacelike under the flow.
\end{corollary}

\begin{proposition}[Short time existence]\label{short time existence MCF}
    Suppose that $\S$ is compact, spacelike-convex and let $F_0: \S^n \to \R^{n+1,k}$ be a smooth immersion, then there exists $\e > 0$ and a smooth solution $F(\cdot,t): \S^n \to \R^{n+1,k}$ to MCF defined for $t\in [0, \e)$ satisfying $F(\cdot,0) = F_0$.
\end{proposition}

\section{Noncollapsing}\label{noncollapsing section}

The result holds for a more general class of submanifolds, which is an analogue of mean-convex hypersurfaces.

\begin{definition}
    A spacelike submanifold $\S^n \subset \R^{n+1,k}$ is \emph{spacelike-mean-convex} if its mean curvature is everywhere spacelike and non-vanishing.
\end{definition}

First, we show that spacelike-mean-convexity is preserved, assuming that the induced metric stays spacelike, this condition is implied by noncollapsing so we omit the assumption for results in this section.
\begin{proposition}
    Suppose that $\S$ is compact, spacelike-mean-convex and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, then spacelike-mean-convexity is preserved along the flow.
\end{proposition}

The assumption of turning number $1$ is carried out throughout the paper. The Euclidean inner product leads to a notion of long time existence, hence a subsequence convergence result that a suitably rescaled flow admits a subsequence that converges smoothly to a sphere in some maximal spacelike subspace $\R^{n+1,0} \subset \R^{n+1,k}$. The approach to full convergence is motivated by first author's approach on Gauss curvature flow with the idea of considering normalised flow invariant under affine transformations \cite{andrews96} and showing that the original flow is stable near the final shape \cite[Chapter 16]{ben2020}. Since the rescaled flow consists of a Lorentz transform at each subsequent time, we consider the normalised flow that fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume. This leads to a stability argument which proves our final result: an analogue of Huisken's theorem.

\begin{theorem}
    Let $F: \S^n \times [0, T) \to \R^{n+1,k}$ be a maximal solution to MCF, if $F(\cdot, 0)$ is a spacelike-convex embedding, then $F(\cdot, t)$ is a spacelike-convex embedding for all $0 < t < T $, $F_t$ converges uniformly to a point as $t \to T$. Moreover, a suitably rescaled flow $\Tilde{F}(\cdot, \tau)$ exists for all $\tau \in [0, \infty)$ and converges in smooth topology to a smooth embedding whose image coincides with $S^n \subset \R^{n+1,0}$ of $\R^{n+1,k}$.
\end{theorem}

For $n = 1$, this is an analogue of Gage--Hamilton theorem.

The post-theorem introduction does not sketch a proof of the stated noncollapsing preservation result. It instead outlines an approach toward a later “final result” (an analogue of Huisken's theorem): using the Euclidean inner product to obtain “long time existence” and a “subsequence convergence result” for a suitably rescaled flow converging smoothly to a sphere in a maximal spacelike subspace; then, to upgrade subsequence convergence to full convergence, it follows an approach “motivated by” work on Gauss curvature flow by considering a “normalised flow invariant under affine transformations” and proving the “original flow is stable near the final shape.” Because the rescaled flow involves “a Lorentz transform at each subsequent time,” it instead considers a normalised flow that “fixes the volume, centre of mass and the maximal spacelike subspace that has the minimum projected volume,” leading to “a stability argument” establishing the Huisken-type convergence.

Suppose that $\S$ is compact, spacelike (mean) convex and acausal, and $F: \S^n \times I \to \R^{n+1,k}$ is a solution to MCF, and for $n = 1$ we require the curve to have turning number (degree of the Gauss map) $1$ , then noncollapsing is preserved.,

Question:
Let $F:\Sigma^n\times I\to\mathbb{R}^{n+1,k}$ be a solution of mean curvature flow $(\partial_tF=H)$ for a compact acausal spacelike submanifold, where acausal means that no two points of the submanifold are timelike-separated in the ambient pseudo-Euclidean space $\mathbb{R}^{n+1,k}$. Assume the submanifold is spacelike (mean) convex: either it is spacelike-convex, meaning $h(v,v)$ is strictly spacelike for every nonzero tangent vector $v$, or it is spacelike-mean-convex, meaning the mean curvature vector $H$ is everywhere spacelike and nonzero. If $n=1$, also assume the curve has turning number (degree of the Gauss map) $1$. For $\delta>0$, say a time-slice is $\delta$-interior noncollapsed if for every $x\in\Sigma$ the pseudoball
$$\left\{p:\left|p-F(x)-\delta\frac{H(x)}{|H(x)|^2}\right|<\frac{\delta}{|H(x)|}\right\}$$
touches the submanifold at $F(x)$ and the image lies outside it; define $\delta$-exterior noncollapsed analogously using the corresponding exterior touching pseudoball. Which statement holds for every such flow?

Noncollapsing is preserved along the flow: if the initial slice is noncollapsed (in particular, if it is $\delta_+$-interior noncollapsed and/or $\delta_-$-exterior noncollapsed), then every later time-slice remains noncollapsed with the same corresponding noncollapsing parameter(s) for all times in $I$.

Choices:
B. Noncollapsing improves along the flow: if the initial slice is noncollapsed, then for every later time-slice there exists a parameter strictly larger than the initial interior noncollapsing parameter and strictly larger than the initial exterior noncollapsing parameter for which the corresponding interior and exterior noncollapsing inequalities hold throughout $I$.
C. One-sided noncollapsing is preserved along the flow: if the initial slice is $\delta_+$-interior noncollapsed, then every later time-slice remains $\delta_+$-interior noncollapsed for all times in $I$; likewise, if the initial slice is $\delta_-$-exterior noncollapsed, then every later time-slice remains $\delta_-$-exterior noncollapsed.
D. Noncollapsing holds automatically along the flow from the geometric assumptions alone: every compact acausal spacelike (mean) convex solution, with the turning-number-$1$ hypothesis when $n=1$, is both $\delta_+$-interior noncollapsed and $\delta_-$-exterior noncollapsed for some positive constants $\delta_+,\delta_-$ depending only on $n$ and $k$, for all times in $I$, without any initial noncollapsing assumption.
E. Noncollapsing is preserved along the flow provided one strengthens the initial hypothesis to two-sided noncollapsing: if the initial slice is simultaneously $\delta$-interior noncollapsed and $\delta$-exterior noncollapsed for the same parameter $\delta>0$, then every later time-slice remains both $\delta$-interior and $\delta$-exterior noncollapsed for all times in $I$; however, preservation need not hold for only one of the two sides separately.