标题： 闵可夫斯基空间中类空超曲面的稳定性和刚性结果 显示英文标题

标题： Stability and rigidity results of space-like hypersurface in the Minkowski space

摘要： 本文利用Weinberger型方法与P函数和积分恒等式，建立了Minkowski空间中类空超曲面的一些刚性定理。 首先，对于以区域$\Omega\subset\mathbb R^n$上的图形$x_{n+1}=u(x)$表示的类空超曲面$M$，如果高阶平均曲率比$\frac{H_{k}}{H_l}(l<k)$为常数且边界$\partial M$位于与之以恒定角度相交的超平面，则该超曲面必为双曲面的一部分。 其次，对于具有边界位于双曲面或光锥上的凸类空超曲面，如果高阶平均曲率比$\frac{H_{k}}{H_l}(l<k)$为常数且超曲面与双曲面（或光锥）在边界处法向量之间的角函数为常数，则此类超曲面必为双曲面的一部分。 这些结果显著扩展了高在\cite{Gao1,Gao2}中的工作。 此外，我们为常平均曲率（CMC）图形超曲面$x_{n+1}=u(x)$、$x\in\Omega\subset\mathbb R^n$得出两个基本的积分恒等式，且其边界位于一个超平面。 作为一些应用：我们通过曲率性质得到了识别双曲面的完全等价条件。 我们还建立了几何稳定性估计，表明 $\bar h$ 的平方范数（ $M$ 的无迹第二基本形式）被 $\partial\Omega$ 的几何量定量控制，如以下不等式所示： $$ ||\bar h||_{L^2(\Omega)}\leq C(n,K)||H_{\partial\Omega}-H_0||_{L^1(\partial\Omega)}^{1/2}. $$ 其中， $H_{\partial\Omega}$ 是 $\partial\Omega$ 的平均曲率， $H_0$ 是某个参考常数， $C$ 是一个常数。 最后，建立了类似的估计。 显示英文摘要

摘要： In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs $x_{n+1}=u(x)$ over domain $\Omega\subset\mathbb R^n$, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the boundary $\partial M$ lies on a hyperplane intersecting with constant angles, then the hypersurface must be a part of hyperboloid. Secondly, for convex space-like hypersurfaces with boundaries on a hyperboloid or light cone, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the angle function between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) on the boundary is constant, then such hypersurfaces must be a part of hyperboloid. These results significantly extend Gao's previous work presented in \cite{Gao1,Gao2}. Furthermore, we derive two fundamental integral identities for constant mean curvature (CMC) graphical hypersurfaces $x_{n+1}=u(x)$, $x\in\Omega\subset\mathbb R^n$, and the boundary lies on a hyperplane. As some applications: we obtain complete equivalence conditions for hyperboloid identification through curvature properties. We also establish a geometric stability estimate demonstrating that the square norm of the trace-free second fundamental form $\bar h$ of $M$ is quantitatively controlled by geometric quantities of $\partial\Omega$, as expressed by the inequality: $$ ||\bar h||_{L^2(\Omega)}\leq C(n,K)||H_{\partial\Omega}-H_0||_{L^1(\partial\Omega)}^{1/2}. $$ Here, $H_{\partial\Omega}$ is the mean curvature of $\partial\Omega$, $H_0$ is some reference constant and $C$ is a constant. Finally, analogous estimates are established.