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.
Categories: math.DG