The polyharmonic heat flow of closed plane curves

The polyharmonic heat flow of closed plane curves

On Partitions of Plane Sets into Simple Closed Curves

We investigate the conjecture that the complement in the euclidean plane E2 of a set F of cardinality less than the continuum c can be partitioned into simple closed curves iff F has a single point. The case in which F is finite was settled in [1] where it was used to prove that, among the compact connected two-manifolds, only the torus and the Klein bottle can be so partitioned. Here we prove ...

Closed Timelike Curves and Holography in Compact Plane Waves

We discuss plane wave backgrounds of string theory and their relation to Gödel–like universes. This involves a twisted compactification along the direction of propagation of the wave, which induces closed timelike curves. We show, however, that no such curves are geodesic. The particle geodesics and the preferred holographic screens we find are qualitatively different from those in the Gödel–li...

Well-posedness for the heat flow of polyharmonic maps with rough initial data

We establish both local and global well-posedness of the heat flow of polyharmonic maps from R to a compact Riemannian manifold without boundary for initial data with small BMO norms.

Convergence of the heat flow for closed geodesics

On a closed Riemannian manifold pM, gq, the geodesic heat flow deforms loops through an energydecreasing homotopy uptq. It is known that there is a sequence tn Ñ 8 so that uptnq converges in C to a closed geodesic, but it is also known that convergence fails in general. We show that for a generic set of metrics on M , the heat flow converges for all initial loops. We also show that convergence,...