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A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $\Delta \vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $\Delta$ is the rough…

For constant mean curvature surfaces of class $C^2$ immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of…

We give the classification of constant mean curvature rotational surfaces of elliptic, hyperbolic, and parabolic type in the four-dimensional pseudo-Euclidean space with neutral metric.

We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let $(M^3,g)$ be an asymptotically flat manifold with scalar curvature $R\ge…

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant…

This paper establishes the conditions under which minimal and stable minimal hypersurfaces are characterized as hyperplanes in Euclidean spaces and as totally geodesic submanifolds in Riemannian manifolds.

In [16] there was proved that any biharmonic hypersurface with at most three distinct principal curvatures in space forms has constant mean curvature. At the very last step of the proof, the argument relied on the fact that the resultant of…

In this paper, we study complete $\delta$-stable minimal hypersurfaces in $\mathbf R^{n+1}$. We prove that complete two-sided $\delta$-stable minimal hypersurfaces have Euclidean volume growth if $3\leq n\leq 5$ and $\delta>\delta_0(n)$,…

We survey our recent results on classifying complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature"-- one is the total absolute curvature which…

In this paper, we classify Euclidean umbilic-free hypersurfaces with semi-parallel Moebius second fundamental form and three distinct principal curvatures. This completes the classification of such hypersurfaces initiated by Hu, Xie and…

In this paper, we study hypersurfaces $M_{r}^{4}$ $(r=0, 1, 2, 3, 4)$ satisfying $\triangle \vec{H}=\lambda \vec{H}$ ($\lambda$ a constant) in the pseudo-Euclidean space $\mathbb{E}_{s}^{5}$ $(s=0, 1, 2, 3, 4, 5)$. We obtain that every such…

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 show that a complete Euclidean submanifold with minimal index of relative nullity $\nu_0>0$ and Ricci curvature with a certain controlled decay must be a $\nu_0$-cylinder. This is an extension of the classical Hartman cylindricity…

We prove a sharp inequality for hypersurfaces in the n-dimensional Anti-deSitter-Schwarzschild manifold for general n greater or equal to 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three…

We revisit Allendoerfer-Weil's formula for the Euler characteristic of embedded hypersurfaces in constant sectional curvature manifolds, first taking some time to re-prove it while demonstrating techniques of [2] and then applying it to…

In this paper, we study self-expanding solutions for mean curvature flows and their relationship to minimal cones in Euclidean space. In [18], Ilmanen proved the existence of self-expanding hypersurfaces with prescribed tangent cones at…

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike…

We study hypersurfaces in the pseudo-Euclidean space $\mathbb{E}^{n+1}_s$, which write as a warped product of a $1$-dimensional base with an $(n-1)$-manifold of constant sectional curvature. We show that either they have constant sectional…

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a K\"ahler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A…

We classify biharmonic submanifolds with certain geometric properties in Euclidean spheres. For codimension 1, we determine the biharmonic hypersurfaces with at most two distinct principal curvatures and the conformally flat biharmonic…