Related papers: A note on the existence of H-bubbles via perturbat…

We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of…

Let $M^n$ be a closed immersed hypersurface lying in a contractible ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st…

This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence…

The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form…

We consider surfaces with parallel mean curvature vector (pmc surfaces) in $\mathbb{C}P^n\times\mathbb{R}$ and $\mathbb{C}H^n\times\mathbb{R}$, and, more generally, in cosymplectic space forms. We introduce a holomorphic quadratic…

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type $$ -div\left(\frac{\nabla u}{\sqrt{1+|\nabla…

We give necessary and sufficient conditions on the curvature and the torsion of a regular curve of the space forms $\h^3$ and $\s^3$ to be contained in a totally umbilical surface. In case that the curve has constant torsion, we obtain the…

In this article, we consider compact surfaces $\Sigma$ having constant mean curvature $H$ ($H$-surfaces) whose boundary $\Gamma=\partial\Sigma\subset \mathbb{M}_0= \mathbb{M} \times_f\{0\}$ is transversal to the slice $\mathbb{M}_0$ of the…

In this paper, we show that any open orientable surface S can be properly embedded in H^3 as a minimizing H-surface for any 0<=H<1. We obtained this result by proving a version of the bridge principle at infinity for H-surfaces. We also…

This article is dedicated to prove Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main theorem states that any hyperbolic sphere with $n$ cusps has a…

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power…

We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial…

We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3…

For $n\geq 2$ we define a notion of umbilicity for hypersurfaces in the Heisenberg group $H_{n}$. We classify umbilic hypersurfaces in some cases, and prove that Pansu spheres are the only umbilic spheres with positive constant $p$(or…

We study almost complex surfaces in the nearly K\"ahler $S^3\times S^3$. We show that there is a local correspondence between almost complex surfaces and solutions of the H-surface equation introduced by Wente. We find a global holomorphic…

We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The solution is either an embedded minimal…

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with…

We develop an analytic theory of existence and regularity of surfaces (given by graphs) arising from the geometric minimization problem $$\min_{\mathcal{M}}\frac{1}{2}\int_{\mathcal{M}}|\nabla_{\mathcal{M}}H|^2\,dA$$ where $\mathcal{M}$…

Generalizing a theorem of Huang, Cheng and Wan classified the complete hypersurfaces of $\mathbb R^4$ with non-zero constant mean curvature and constant scalar curvature. In our work, we obtain results of this nature in higher dimensions.…

In this paper, we first give some new characterizations of geodesic spheres in the hyperbolic space by the condition that hypersurface has constant weighted shifted mean curvatures, or constant weighted shifted mean curvature ratio, which…