The Mean Curvature Flow for Isoparametric Submanifolds

RICCI CURVATURE OF SUBMANIFOLDS OF A SASAKIAN SPACE FORM

Involving the Ricci curvature and the squared mean curvature, we obtain basic inequalities for different kind of submaniforlds of a Sasakian space form tangent to the structure vector field of the ambient manifold. Contrary to already known results, we find a different necessary and sufficient condition for the equality for Ricci curvature of C-totally real submanifolds of a Sasakian space form...

The Mean Curvature Flow Smoothes Lipschitz Submanifolds

The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of 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 ...

Mean Curvature Flow of Pinched Submanifolds to Spheres

The evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of Gerhard Huisken’s seminal paper [Hu1]. More recently, mean curvature flow of higher codimension submanifolds has also received attention. In this paper we prove a result analogous to that of [Hu1] for submanifolds of any codimension. Let F0 : Σn → Rn+k be a smooth immersion of a compa...

Mean Curvature Flow of Higher Codimension in Hyperbolic Spaces

where H(x, t) is the mean curvature vector of Ft(M) and Ft(x) = F (x, t). We call F : M × [0, T ) → F(c) the mean curvature flow with initial value F . The mean curvature flow was proposed by Mullins [17] to describe the formation of grain boundaries in annealing metals. In [3], Brakke introduced the motion of a submanifold by its mean curvature in arbitrary codimension and constructed a genera...

Shape Operator of Slant Submanifolds in Kenmotsu Space Forms