Related papers: Stable minimal hypersurfaces in $\mathbf{R}^5$

In a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel, defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces…

We prove that nonnegative $3$-intermediate Ricci curvature combined with uniformly positive $k$-triRic curvature implies rigidity of complete noncompact two-sided stable minimal hypersurfaces in a Riemannian manifold $(X^5,g)$ with bounded…

Stable compact minimal submanifolds of the product of a sphere and any Riemannian manifold are classified whenever the dimension of the sphere is at least three. The complete classification of the stable compact minimal submanifolds of the…

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…

We prove that there are no minimal hypersurfaces properly immersed in any region of the Euclidean space bounded by unstable minimal cones. We also prove the analogous result for $r$-minimal hypersurfaces.

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…

For $s \in (0,1)$ small, we show that the only cones in $\mathbb{R}^2$ stationary for the $s$-perimeter and stable in $\mathbb{R}^2 \setminus \{0\}$ are half-planes. This is in direct contrast with the case of the classical perimeter or the…

We study the global behavior of (weakly) stable constant mean curvature hypersurfaces in general Riemannian manifolds. By using harmonic function theory, we prove some one-end theorems which are new even for constant mean curvature…

We construct examples of flat surfaces in $\mathbb{H}^3$ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in $\mathbb{H}^3$ with only one end and at most two isolated singularities.

A surface of constant mean curvature (CMC) equal to $H$ in a sub-Riemannian $3$-manifold is strongly stable if it minimizes the functional $\text{area}+2H\,\text{volume}$ up to second order. In this paper we obtain some criteria ensuring…

We prove that half spaces are the only stable nonlocal $s$-minimal cones in $\mathbb{R}^3$, for $s\in(0,1)$ sufficiently close to $1$. This is the first classification result of stable $s$-minimal cones in dimension higher than two. Its…

In this paper, we study stable weighted minimal hypersurfaces in manifolds with nonnegative Bakry-Emery Ricci curvature. We will give some geometric and topological applications. In particular, we give some partial classification of…

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal…

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…

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and is used to determine the conformally flat metric $f^{-2}\delta_{ij}$ on the Euclidean space…

The rigidity of the positive mass theorem states that the only complete asymptotically flat manifold of nonnegative scalar curvature and zero mass is Euclidean space. We prove a corresponding stability theorem for spaces that can be…

In this paper, we study four dimensional hypersurface M^4_r with proper mean curvature vector field (i.e. \Delta\vec{H} is proportional to \vec{H}) in pseudo-Riemannian space form N^5_s(c), and show that it has constant mean curvature, and…

We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.

Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $\mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $\left( \int _ { M } | H | ^ { n } d v…

We construct a weakly complete flat surface in hyperbolic 3-space having a pair of hyperbolic Gauss maps both of whose images are contained in an arbitrarily given open disc in the ideal boundary of H^3. This construction is accomplished as…