arXiv subjects

Knut Smoczyk

Publications and source records attributed to Knut Smoczyk.

23 records · Page 2Linked to original sources

arXiv2005

Self-shrinkers of the mean curvature flow in arbitrary codimension

For hypersurfaces of dimension greater than one, Huisken showed that compact self-shrinkers of the mean curvature flow with positive scalar mean curvature are spheres. We will prove the following extension: A compact self-similar solution in arbitrary codimension and of dimension greater than one is spherical, i.e. contained in a sphere, if and only if the mean curvature vector \be H\ee is non-vanishing and the principal normal \beν\ee is parallel in the normal bundle. We also give a classification of complete noncompact self-shrinkers of that type.

math.DG↗

arXiv2004

Mean curvature flow with flat normal bundles

We show that flatness of the normal bundle is preserved under the mean curvature flow in the Euclidean space and use this to generalize a classical result for hypersurfaces due to Ecker-Huisken in the case of submanifolds with arbitrary codimension.

math.DG↗

arXiv2002

Mean Curvature Flows of Lagrangian Submanifolds with Convex Potentials

This article studies the mean curvature flow of Lagrangian submanifolds. In particular, we prove the following global existence and convergence theorem: if the potential function of a Lagrangian graph in T^{2n} is convex, then the flow exists for all time and converges smoothly to a flat Lagrangian submanifold.

math.DG↗

arXiv1996

A canonical way to deform a Lagrangian submanifold

We derive some important geometric identities for Lagrangian submanifolds immersed in a Kähler manifold and prove that there exists a canonical way to deform a Lagrangian submanifold by a parabolic flow through a family of Lagrangian submanifolds if the ambient space is a Ricci-flat Calabi-Yau manifold.

dg-ga↗

arXiv1996

The Fermi Flow and its Application to Geometry

We introduce the notion of Fermi flow for hypersurfaces in Riemannian manifolds. It turns out that this is a powerful tool to study the geometry of distance surfaces about a given initial hypersurface. Some of the results in this paper are known in one form or another, however the aim is to demonstrate how they can all be derived by the same method and proved in a very simple manner. In addition we obtain some new results and results that are stronger than those stated in the literature.

dg-ga↗