Related papers: A Heintze-Karcher type inequality in hyperbolic sp…
In this paper, we consider a fully nonlinear curvature flow of a convex hypersurface in the Euclidean n-space. This flow involves k-th elementary symmetric function for principal curvature radii and a function of support function. Under…
In this paper, we prove an optimal isoperimetric inequality for spacelike, compact, star-shaped, and $2$-convex hypersurfaces in de Sitter space.
We extend to Minkowski spaces the classical result of Barbosa and do Carmo [1] that characterizes the euclidean sphere as the unique compact stable CMC hypersurface of $\mathbb R^n$. More precisely, if $K$ is a smooth convex body in…
A numerical scheme is developed for solution of the Goursat problem for a class of nonlinear hyperbolic systems with an arbitrary number of independent variables. Convergence results are proved for this difference scheme. These results are…
We consider the problem $F=f(\nu)$ for strictly convex, closed hypersurfaces in $S^{n+1}$ and solve it for curvature functions $F$ the inverses of which are of class $(K)$.
In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that…
We prove that a connected mean convex region in $\mathbb{R}^{n+1}$ with at least two components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature…
In this paper, the long-time existence and convergence results are derived for locally constrained flows with initial value some compact spacelike hypersurface that is suitably pinched in the de Sitter space. As applications, geometric…
A surface in hyperbolic space $\h^3$ invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of $\h^3$ that satisfy a linear Weingarten relation of the form…
We investigate the problem of finding smooth hypersurfaces of constant mean curvature in hyperbolic space, which can be represented as radial graphs over a subdomain of the upper hemisphere. Our approach is variational and our main results…
In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially…
We prove a differential Harnack inequality for noncompact convex hypersurfaces flowing with normal speed equal to a symmetric function of their principal curvatures. This extends a result of Andrews for compact hypersurfaces. We assume that…
In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic…
We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in…
We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem…
In this paper, we give some new characterizations of umbilic hypersurfaces in general warped product manifolds, which can be viewed as generalizations of the work in \cite{KLP18} and \cite{WX14}. Firstly, we prove the rigidity for…
In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4 pi are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces…
We show that given a quasi-circle $C$ in $\partial_{\infty}\mathbb{H}^3$ (respectively in $\partial_{\infty} \mathbb{ADS}^3$) and a complete conformal metric $h$ on $\mathbb{D}$ whose curvature $K_h$ takes values in a compact subset of…
We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains…
In this paper, we establish a curvature estimate for semi-convex solutions of Hessian equations in hyperbolic space. We also obtain a curvature estimate for admissible solutions to prescribed curvature measure type problem in hyperbolic…