Related papers: Asymptotic Dirichlet problems in warped products
In this paper we consider a set $E\subset\Omega$ with prescribed mean curvature $f\in C(\Omega)$ and Euclidean Lipschitz boundary $\partial E=\Sigma$ inside a three-dimensional contact sub-Riemannian manifold $M$. We prove that if $\Sigma$…
Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum…
In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $\Omega$ and $\tilde{\Omega}$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there…
In this article, we give results of prescribing scalar and mean curvature functions for metrics either pointwise conformal or conformally equivalent to a Riemannian metric that is equipped on a compact manifold with boundary, with…
We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form…
In this paper, we mainly investigate the set of critical points associated to solutions of mean curvature equation with zero Dirichlet boundary condition in a strictly convex domain and a nonconvex domain respectively. Firstly, we deduce…
We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold $M\times\mathbb{R}$, where $M$ is asymptotically flat. If the initial hypersurface $F_0\subset M\times\mathbb{R}$…
We consider the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature on a compact manifold with boundary, and establish a necessary and sufficient…
The present article provides a study of $2-$Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson-Walker space-times. Some conditions for a $2-$Killing…
We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a…
We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…
We further study averaged and firmly nonexpansive mappings in the setting of geodesic spaces with a main focus on the asymptotic behavior of their Picard iterates. We use methods of proof mining to obtain an explicit quantitative version of…
We establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in Riemannian manifold $(N^{n+1}, \bar g)$ through regularity study of a degenerate fully nonlinear curvature equation in general…
In this paper, we study the Dirichlet problem for the minimal surface equation in $\rm Sol_3$ with possible infinite boundary data, where $\rm Sol_3$ is the non-abelian solvable $3$-dimensional Lie group equipped with its usual…
We prove the existence of branched immersed constant mean curvature 2-spheres in an arbitrary Riemannian 3-sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3-sphere is positively…
We study that the $n-$graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$…
In this paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This…
We express characteristic numbers of compact hyperk\"ahler manifolds in graph-theoretical form, considering them as a special case of the curvature invariants introduced by Rozansky and Witten. The appropriate graphs are generated by…
In this paper we consider the prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds. By deducing the expression of the Gauduchon scalar curvature under the conformal variation, the problem is reduced to solve a…
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…