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

Abstract

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: $$||\bar{h}|{|}_{L^2(\Omega )}\le C(n,K)||H_{\partial \Omega }-H_0|{|}_{L^1(\partial \Omega )}^{1/2}.$$ 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.