A pinching problem on submanifolds with parallel mean curvature vector field in a sphere

Submanifolds with Parallel Mean Curvature Vector in Pinched Riemannian Manifolds

In this paper, we prove a generalized integral inequality for submanifolds with parallel mean curvature vector in an arbitrary Riemannian manifold, and from which we obtain a pinching theorem for compact oriented submanifolds with parallel mean curvature vector in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by Alencar-do Carmo and Hong-Wei Xu.

On the mean curvature of a unit vector field. ∗

On the mean curvature of a unit vector field. Abstract We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an example of a vector field of constant mean curvature on the Lobachevsky (n + 1) space.

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Constant mean curvature surfaces with boundary on a sphere

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must be spherical. Our results apply in many scenarios in physics where in absence...

Neck Pinching Dynamics Under Mean Curvature Flow

In this paper we study motion of surfaces of revolution under the mean curvature flow. For an open set of initial conditions close to cylindrical surfaces we show that the solution forms a “neck” which pinches in a finite time at a single point. We also obtain a detailed description of the neck pinching process.