Related papers: Phase transitions and minimal hypersurfaces in hyp…
In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces $F:M^n\rightarrow\mathbb{R}^{n+1}$ with free boundary on the standard unit sphere. First we show that if $F$ is graphical…
Unlike the case of surfaces of topologically finite type, there are several different Teichm\"uller spaces that are associated to a surface of topological infinite type. These Teichm\"uller spaces first depend (set-theoretically) on whether…
In a recent paper Jorge and Mercuri proved that the image of Gauss map of a complete non flat minimal surfaces in R3 with finite total curvature omits at most 2 points. In this work we follow their idea and prove 3a similar result for CMC-1…
In this paper, we study the asymptotic Plateau problem in hyperbolic space for constant sum Hessian curvature. More precisely, given a asymptotic boundary $\Gamma$, one seeks a complete hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ satisfying…
In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface $M$ in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on $M$. We provide some…
We prove that any weakly acausal curve $\Gamma$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$.…
We investigate minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic graph. Under some natural conditions on the graph, we show the existence of non-constant uniformly-bounded minimal solutions with prescribed asymptotic…
We study translation minimal hypersurfaces and separable minimal hypersurfaces in the ($n+1$)-space with $2m$-norm.
We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of…
We construct new examples of embedded, complete minimal hypersurfaces in quaternionc hyperbolic space and also some minimal foliations. We introduce fans an construct analytic deformations of bisectors.
We give several characterizations of parabolic (quasisuper)- minimizers in a metric measure space equipped with a doubling measure and supporting a Poincar\'e inequality. We also prove a version of comparison principle for super- and…
Let N be a complete, simply-connected surface of constant curvature \kappa \leq 0. Moreover, suppose that \Omega and \tilde{\Omega} are strictly convex domains in N with the same area. We show that there exists an area-preserving…
On a closed Riemannian surface of negative curvature, we prove a characterization for configurations of closed geodesics arising from one parameter Allen-Cahn min-max constructions. One of the facts we conclude is that every geodesic occurs…
We find complete hypersurfaces of constant curvature in hyperbolic space with a prescribed asymptotic boundary at infinity for a general class of (elliptic) curvature functions which includes the higher order mean curvatures and their…
In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial…
Given a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ tends to zero as $j$…
We prove that if an $n$-dimensional complete minimal submanifold $M$ in hyperbolic space has sufficiently small total scalar curvature then $M$ has only one end. We also prove that for such $M$ there exist no nontrivial $L^2$ harmonic…
A boundary value problem related to a parabolic higher order operator with a small parameter is analized. When the small parameter tends to zero, the reduced operator is hyperbolic. When t tends to infinity a parabolic hyperbolic boundary…
An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn-Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward…
In this paper we study the common distance between points and the behavior of a constant length step discrete random walk on finite area hyperbolic surfaces. We show that if the second smallest eigenvalue of the Laplacian is at least 1/4,…