Related papers: A geometric inequality on hypersurface in hyperbol…
In this work, we show equidistribution properties for the horocycles of a geometrically finite surface with variable negative curvature. If the surface is hyperbolic, we deduce an equidistribution result for the orbits of the horocyclic…
We describe the structure of the singular sets of constant curvature, convex hypersurfaces in hyperbolic space for general convex curvature functions. We apply this result to the study of the ideal Plateau problem in hyperbolic space for…
In this paper, we investigate the prescribed total geodesic curvature problem for generalized circle packing metrics in hyperbolic background geometry on surfaces with infinite cellular decompositions. To address this problem, we introduce…
We study a new notion of convexity for subsets of the unit sphere, which closely resembles the horo-convexity for subsets of the hyperbolic space. We call this notion, accordingly, horo-convexity. For horo-convex hypersurfaces of the unit…
Brendle [6] successfully establishes the sharp Michael-Simon inequality for mean curvature on Riemannian manifolds with nonnegative sectional curvature ($\mathcal{K} \geq 0$), and the proof relies on the Alexandrov-Bakelman-Pucci method.…
We prove a hyperplane inequality for the surface area of projection bodies.
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 propose a Lie geometric point of view on flat fronts in hyperbolic space as special omega-surfaces and discuss the Lie geometric deformation of flat fronts.
We consider convex, spacelike hypersurfaces with boundaries on some hyperboloid (or lightcone) in the Minkowski space. If the hypersurface has constant higher order mean curvature, and the angle between the normal vectors of the…
We consider contracting flows in $(n+1)$-dimensional hyperbolic space and expanding flows in $(n+1)$-dimensional de Sitter space. When the flow hypersurfaces are strictly convex we relate the contracting hypersurfaces and the expanding…
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…
In this paper, we study the power of Gaussian curvature flow of a compact convex hypersurface and establish its Harnack inequality when the power is negative. In the Harnack inequality, we require that the absolute value of the power is…
We classify all real hypersurfaces with three distinct constant principal curvatures in complex hyperbolic spaces of dimension greater than two.
We consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. We prove that the ratio of circumradius to inradius is bounded…
We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions,…
We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle…
In this paper, we prove a family of sharp geometric inequalities for free boundary hypersurfaces in a ball in space forms.
The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form…
In this article, we first introduce the quermassintegrals for compact hypersurfaces with capillary boundaries in hyperbolic space from a variational viewpoint, and then we solve an isoperimetric type problem in hyperbolic space. By…
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…