We consider closed orientable hypersurfaces in a wide class of warped product manifolds which include space forms, deSitter-Schwarzschild and Reissner-Nordstr\"{o}m manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These results can be viewed as generalizations of the classical Jellet-Liebmann theorem and the Alexandrov theorem in Euclidean space. In particular, Corollary 1.8 implies that the embeddedness condition in Theorem 2 of [Brendle-Eichmair, JDG. 2013] is not necessary.

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regularity theorem of Ecker and Huisken for Lipschitz hypersurfaces. In particular, any submanifold of the Euclidean space with a continuous induced metric can be smoothed out by the mean curvature flow. The smallness assumption is necessary in the higher codimension case in view of an example of Lawson and Osserman. The stationary phase of the mean curvature flow corresponds to minimal submanifolds. Our result thus generalizes Morrey’s classical theorem on the smoothness of $C_1$ minimal submanifolds.

In this paper, we establish a Bochner-type formula on Alexandrov spaces with Ricci curvature bounded below. Yau’s gradient estimate for harmonic functions is also obtained on Alexandrov spaces.

We determine all helix surfaces with parallel mean curvature vector field which are not minimal or pseudo-umbilical in spaces of type $M^{n}(c)\times\mathbb{R}$ , where$M$^{$n$}($c$) is a simply connected$n$-dimensional manifold with constant sectional curvature$c$.

We study the properties of the n-volumic scalar curvature in this note. Lott-Sturm-Villani's curvature-dimension condition CD(κ,n) was showed to imply Gromov's n-volumic scalar curvature ≥nκ under an additional n-dimensional condition and we show the stability of n-volumic scalar curvature ≥κ with respect to smGH-convergence. Then we propose a new weighted scalar curvature on the weighted Riemannian manifold and show its properties.