Curvature estimates for spacelike graphic hypersurfaces in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$

In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the (n + 1)-dimensional Lorentz-Minkowski space R 1 , and through which the existence of spacelike admissible graphic hypersurfaces, with prescribed 2-th Weingarten curvature and Dirichlet boundary data, defined over a strictly convex domain in the hyperbolic plane H (1) ⊂ R 1 of center at origin and radius 1, can be proven.

[1]  L Rafael,et al.  DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES IN LORENTZ-MINKOWSKI SPACE , 2014 .

[2]  R. Bartnik.  Existence of maximal surfaces in asymptotically flat spacetimes , 1984 .

[3]  Philippe Delanoë.  Dirichlet problem for the equation of a given Lorentz-Gaussian curvature , 1990 .

[4]  Xu-jia Wang,et al.  Interior curvature bounds for a class of curvature equations , 2004 .

[5]  Jing Mao,et al.  Inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ , 2021, 2104.10600.

[6]  Ya Gao,et al.  An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane $\mathbb{R}^{2}_{1}$ , 2021 .

[7]  Bo Guan.  The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature , 1998 .

[8]  Ben Andrews,et al.  Contraction of convex hypersurfaces in Euclidean space , 1994 .

[9]  Yong Huang.  Curvature estimates of hypersurfaces in the Minkowski space , 2013 .

[10]  Pengfei Guan,et al.  The Christoffel-Minkowski problem I: Convexity of solutions of a Hessian equation , 2003 .

[11]  J. Spruck,et al.  Nonlinear Second-Order Elliptic Equations V. The Dirichlet Problem for Weingarten Hypersurfaces , 1988 .

[12]  Jing Mao,et al.  A stability result for translating spacelike graphs in Lorentz manifolds , 2021, Acta Mathematica Scientia.

[13]  L. Simon,et al.  Spacelike hypersurfaces with prescribed boundary values and mean curvature , 1982 .

[14]  Ya Gao,et al.  The Dirichlet problem for a class of Hessian quotient equations in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$ , 2021, 2111.02028.