160 research outputs found
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 \be\nu\ee is parallel in the
normal bundle. We also give a classification of complete noncompact
self-shrinkers of that type.Comment: 19 pages, 1 figur
Mean curvature flow of monotone Lagrangian submanifolds
We use holomorphic disks to describe the formation of singularities in the
mean curvature flow of monotone Lagrangian submanifolds in .Comment: 37 pages, 3 figure
The hyperbolic mean curvature flow
We introduce a geometric evolution equation of hyperbolic type, which governs
the evolution of a hypersurface moving in the direction of its mean curvature
vector. The flow stems from a geometrically natural action containing kinetic
and internal energy terms. As the mean curvature of the hypersurface is the
main driving factor, we refer to this model as the hyperbolic mean curvature
flow (HMCF). The case that the initial velocity field is normal to the
hypersurface is of particular interest: this property is preserved during the
evolution and gives rise to a comparatively simpler evolution equation. We also
consider the case where the manifold can be viewed as a graph over a fixed
manifold. Our main results are as follows. First, we derive several balance
laws satisfied by the hypersurface during the evolution. Second, we establish
that the initial-value problem is locally well-posed in Sobolev spaces; this is
achieved by exhibiting a convexity property satisfied by the energy density
which is naturally associated with the flow. Third, we provide some criteria
ensuring that the flow will blow-up in finite time. Fourth, in the case of
graphs, we introduce a concept of weak solutions suitably restricted by an
entropy inequality, and we prove that a classical solution is unique in the
larger class of entropy solutions. In the special case of one-dimensional
graphs, a global-in-time existence result is established.Comment: 26 page
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