Related papers: Strongly stable surfaces in sub-Riemannian $3$-spa…
We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work…
This paper develops a technique for applying one-parameter prescribed mean curvature min-max theory in certain non-compact manifolds. We give two main applications. First, fix a dimension $3\le n+1 \le 7$ and consider a smooth function…
We study constant mean curvature surfaces in the three-dimensional Heisenberg group. We prove that a constant mean curvature surface in a neighborhood of non-umbilic point is described by some solution of a sinh-Gordon equation subject to a…
We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface…
We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci…
We prove that a spacelike spherical symmetric constant mean curvature (SSCMC) surface and a general spacelike constant mean curvature (CMC) surface with certain boundary condition at the future null-infinity in Schwarzschild spacetime are…
In this paper I study the constant mean curvature surface in asymptotically flat 3-manifolds with general asymptotics. Under some weak condition, I prove that outside some compact set in the asymptotically flat 3-manifold with positive…
We give a global version of the Bryant representation of surfaces of constant mean curvature one (cmc-1) in hyperbolic space. This allows to set the associated non-abelian period problem in the framework of flat unitary vector bundles on…
We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in…
In this paper, we study the stability of catenoids and helicoids in the hyperbolic $3$-space $\mathbb{H}^3$. (1) For a family of spherical minimal catenoids $\{\mathcal{C}_a\}_{a>0}$ in $\mathbb{H}^3$, there exist two constants $0<a_c<a_l$…
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…
Even though it is known that there exist quasi-Fuchsian hyperbolic three-manifolds that do not admit any monotone foliation by constant mean curvature (CMC) surfaces, a conjecture due to Thurston asserts the existence of CMC foliations for…
A curve $C$ on a variety $X$ is stably balanced if the slopes of the Harder-Narasimhan filtration of its normal bundle $N$ are contained in an interval of length 1. For each $d\geq n+1$ we construct some regular families of pairs $(C, X)$…
Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kr\"{o}ncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we…
In this article we prove that all boundary points of a minimal oriented hypersurface in a Riemannian manifold are regular, that is, in a neighborhood of any boundary point, the minimal surface is a $\mathcal{C}^{1, \frac14}$ submanifold…
We classify constant mean curvature surfaces invariant by a 1-parameter group of isometries in the Berger spheres and in the special linear group Sl(2, R). In particular, all constant mean curvature spheres in those spaces are described…
Let $(M,\bar{g}, e^{-f}d\mu)$ be a complete metric measure space with Bakry-\'Emery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface…
In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riamannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the stability index of the…
Given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce…
In this paper we study triharmonic hypersurfaces immersed in a space form $N^{n+1}(c)$. We prove that any proper CMC triharmonic hypersurface in the sphere $\mathbb S^{n+1}$ has constant scalar curvature; any CMC triharmonic hypersurface in…