Related papers: One Dimensional Asymptotic Plateau Problem in $n$-…

We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a…

We show the existence of a complete, strictly locally convex hypersurface within $\mathbb{H}^{n+1}$ that adheres to a curvature equation applicable to a broad range of curvature functions. This hypersurface possesses a prescribed asymptotic…

We investigate on the existence of smooth complete hypersurface with prescribed Weingarten curvature and asymptotic boundary at infinity in hyperbolic space under the assumption that there exists an asymptotic subsolution. We give an…

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…

We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at…

We define and prove the existence of unique solutions of an asymptotic Plateau problem for spacelike maximal surfaces in the pseudo-hyperbolic space of signature (2, n): the boundary data is given by loops on the boundary at infinity of the…

We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any…

In this paper, we consider the asymptotic $\sigma_k$ Plateau problem in hyperbolic space. We establish $C^2$ estimates for semi-convex complete hypersurfaces satisfying constant $\sigma_k$ curvature with a prescribed asymptotic boundary at…

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,\tau)$. As one of our main results, we present sufficient conditions for a curve $\Gamma$ in $\partial_{\infty}…

In this article we extend several foundational results of the theory of complete minimal surfaces of finite index in the Euclidean space to minimal surfaces in asymptotically flat manifolds and, more generally, to marginally outer-trapped…

Let $\mathcal{M}$ be a smooth, closed and connected manifold of dimension $n\in\mathbb{N}$, endowed with a Riemannian metric $g$. Moreover, let $\mathcal{B}$ be an $(n+1)$-dimensional compact manifold with boundary equal to $\mathcal{M}$.…

Let (M,g) be a smooth connected compact Riemannian manifold of finite dimension n \geq 2 with a smooth boundary \partial M. We consider the problem -{\epsilon}^2\Delta_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partial{\nu}=0 on \partial M…

We consider a Plateau problem in codimension $1$ in the non-parametric setting. A Dirichlet boundary datum is given only on part of the boundary $\partial \Omega$ of a bounded convex domain $\Omega\subset\mathbb{R}^2$. Where the Dirichlet…

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-\alpha$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then,…

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in…

In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics…

Asymptotic expansion is constructed and justified for the solution to a nonuniform Neumann boundary-value problem for the Poisson equation with the right-hand side that depends both on longitudinal and transversal variables in a thin…

We establish a lower bound for the number of sign changing solutions with precisely two nodal domains to the singularly perturbed nonlinear elliptic equation -{\epsilon}^{2}{\Delta}_{g}u+u=|u|^{p-2}u on an n-dimensional Riemannian manifold…

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Amp\`ere equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and…

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin $3D$ aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter $\mathcal{O}(\varepsilon).$ A…