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

TL;DR

This paper proves new rigidity theorems for space-like hypersurfaces in Minkowski space, characterizing hyperboloids via curvature ratios and boundary conditions, and derives integral identities and stability estimates for such hypersurfaces.

Contribution

It introduces novel rigidity results for space-like hypersurfaces with constant curvature ratios, extending previous work, and provides integral identities and stability estimates related to curvature properties.

Findings

Hypersurfaces with constant higher-order mean curvature ratios are hyperboloids.

Boundary conditions on hyperplanes, hyperboloids, or light cones determine hypersurface shape.

A stability estimate links the trace-free second fundamental form to boundary curvature differences.

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