Stability and rigidity results of space-like hypersurface in the Minkowski space
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2025
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| author | Chen, Jianhua Deng, Haiyun Xie, Haiqin Yin, Jiabin |
| author_facet | Chen, Jianhua Deng, Haiyun Xie, Haiqin Yin, Jiabin |
| contents | 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 $Ω\subset\mathbb R^n$, 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 $x_{n+1}=u(x)$, $x\inΩ\subset\mathbb R^n$, 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 $\partialΩ$, as expressed by the inequality: $$ ||\bar h||_{L^2(Ω)}\leq C(n,K)||H_{\partialΩ}-H_0||_{L^1(\partialΩ)}^{1/2}. $$
Here, $H_{\partialΩ}$ is the mean curvature of $\partialΩ$, $H_0$ is some reference constant and $C$ is a constant.
Finally, analogous estimates are established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_01012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability and rigidity results of space-like hypersurface in the Minkowski space Chen, Jianhua Deng, Haiyun Xie, Haiqin Yin, Jiabin Differential Geometry 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 $Ω\subset\mathbb R^n$, 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 $x_{n+1}=u(x)$, $x\inΩ\subset\mathbb R^n$, 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 $\partialΩ$, as expressed by the inequality: $$ ||\bar h||_{L^2(Ω)}\leq C(n,K)||H_{\partialΩ}-H_0||_{L^1(\partialΩ)}^{1/2}. $$ Here, $H_{\partialΩ}$ is the mean curvature of $\partialΩ$, $H_0$ is some reference constant and $C$ is a constant. Finally, analogous estimates are established. |
| title | Stability and rigidity results of space-like hypersurface in the Minkowski space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.01012 |