相关论文: Strictly Locally Convex Hypersurfaces with Prescri…

Anatomical structures such as the hippocampus, liver, and bones can be analyzed as orientable, closed surfaces. This permits the computation of volume, surface area, mean curvature, Gaussian curvature, and the Euler-Poincar\'e…

We establish a concavity principle for solutions to elliptic and parabolic equations on locally symmetric spaces with nonnegative sectional curvature, extending the results of Langford and Scheuer. To the best of our knowledge, this is the…

We give a description (direct formulas) of all complex geodesics in a convex tube domain in $\CC^n$ containing no complex affine lines, expressed in terms of geometric properties of the domain. We next apply that result to give formulas (a…

Motivated by previous study on mean curvature flow and prescribed mean curvature flow on spatially compact space or asymptotically flat spacetime, in this work we will find sufficient conditions for the short time existence of prescribed…

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bounds on the…

We consider inverse curvature flows in warped product manifolds, which are constrained subject to local terms of lower order, namely the radial coordinate and the generalized support function. Under various assumptions we prove longtime…

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in…

We investigate the minimal and isoperimetric surface problems in a large class of sub-Riemannian manifolds, the so-called Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of…

How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete…

This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean…

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical…

We study the geometry of the foliation by constant Gaussian curvature surfaces $(\Sigma_k)_k$ of a hyperbolic end, and how it relates to the structures of its boundary at infinity and of its pleated boundary. First, we show that the…

Our first objective in this paper is to give a natural formulation of the Christoffel problem for hypersurfaces in $H^{n+1}$, by means of the hyperbolic Gauss map and the notion of hyperbolic curvature radii for hypersurfaces. Our second…

This article is based on a lecture at the Journal of Differential Geometry Conference, Harvard 2017. We discuss closed and torsion-free $G_{2}$-structures on a 7-manifold with boundary, with prescribed $3$-form on the boundary. Much of the…

In a recent article, a localization of the Huisken--Stampacchia iteration method was developed, and used to establish localizations of the well-known "umbilic", "convexity" and "cylindrical" estimates for hypersurfaces evolving in Euclidean…

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In…

We establish a prior interior $C^{1,1}$ estimates for convex solutions and supercritical phase solutions to the Lagrangian mean curvature equation with sharp Lipschitz phase. Counter-examples exist when the phase is H\"{o}lder continuous…

In many applications it is important to establish if a given topological preordered space has a topology and a preorder which can be recovered from the set of continuous isotone functions. Under antisymmetry this property, also known as…

We prove the local convergence of uniform bipartite maps with prescribed face degrees in the high genus regime. Unlike in the previous work arxiv:2012.05813 on the subject, we do not make any assumption on the tail of the face degrees,…

In this overview report we generalize Erhard Heinz' curvature estimate for minimal graphs in R^3 to graphs in R^n of prescribed mean curvature. Secondly, we analyse these problems in the frame of the outer differential geometry which leads…