Related papers: Asymptotic Plateau Problem in H2xR

We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger…

We prove that a H-surface M in H^2xR, |H| <= 1/2, inherits the symmetries of its boundary when the boundary is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one,…

We prove a generalization of Reifenberg's isoperimetric inequality. The main result of this paper is used to establish existence of a minimizer for an anisotropically-weighted area functional among a collection of surfaces which satisfies a…

In this paper we consider the H\'enon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent $p$ is close to…

Classically, Plateau's problem asks to find a surface of the least area with a given boundary $B$. In this article, we investigate a version of Plateau's problem, where the boundary of an admissible surface is only required to partially…

Solving the Plateau problem means to find the surface with minimal area among all surfaces with a given boundary. Part of the problem actually consists of giving a suitable definition to the notions of 'surface', 'area' and 'boundary'. In…

In this paper, we solve the asymptotic Plateau problem in hyperbolic space for constant $\sigma_{n-1}$ curvature, i.e. the existence of a complete hypersurface in $\mathbb{H}^{n+1}$ satisfying $\sigma_{n-1}(\kappa)=\sigma\in (0,n)$ with a…

A general study of minimal surfaces of the Riemannian product of two spheres S^2xS^2 is tackled. We stablish a local correspondence between (non-complex) minimal surfaces of S^2xS^2 and certain pair of minimal surfaces of the sphere S^3.…

Plateau's soap film problem is to find a surface of least area spanning a given boundary. We begin with a compact orientable $(n-2)$-dimensional submanifold $M$ of $\R^n$. If $M$ is connected, we say a compact set $X$ "spans" $M$ if $X$…

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. We show that any minimizing H-disk in an H_0-convex domain is embedded for any H in [0,H_0). In particular, for…

We give a bound, linear in the complexity of the surface, on the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.

We investigate the minimal surface problem in the three dimensional Heisenberg group, H, equipped with its standard Carnot-Caratheodory metric. Using a particular surface measure, we characterize minimal surfaces in terms of a sub-elliptic…

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 prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description…

It is extended a result due to B. Guan and J. Spruck on the asymptotic Plateau's problem for CMC radial graphs in hyperbolic space to horizontal CMC graphs.

We look for minimal conditions on a two-dimensional metric surface $X$ of locally finite Hausdorff $2$-measure under which $X$ admits an (almost) parametrization with good geometric and analytic properties. Only assuming that $X$ is locally…

Let $(M,g)$ be an asymptotically conical Riemannian manifold having dimension $n\ge 2$, opening angle $\alpha \in (0,\pi/2) \setminus \{\arcsin \frac{1}{2k+1}\}_{k \in \mathbb{N}}$ and positive asymptotic rate. Under the assumption that the…

We study asymptotic behaviors of solutions $f$ to the Dirichlet problem for minimal graphs in the hyperbolic space with singular asymptotic boundaries under the assumption that the boundaries are piecewise regular with positive curvatures.…

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz…

We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen-Cahn equation is small enough, there exist minimal…