6,745 research outputs found
Uniqueness and nonuniqueness for Ricci flow on surfaces: Reverse cusp singularities
We extend the notion of what it means for a complete Ricci flow to have a
given initial metric, and consider the resulting well-posedness issues that
arise in the 2D case. On one hand we construct examples of nonuniqueness by
showing that surfaces with cusps can evolve either by keeping the cusps or by
contracting them. On the other hand, by adding a noncollapsedness assumption
for the initial metric, we establish a uniqueness result
Ricci flows with unbounded curvature
We show that any noncompact Riemann surface admits a complete Ricci flow
g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).Comment: 12 pages, 1 figure; updated reference
Flowing maps to minimal surfaces
We introduce a flow of maps from a compact surface of arbitrary genus to an
arbitrary Riemannian manifold which has elements in common with both the
harmonic map flow and the mean curvature flow, but is more effective at finding
minimal surfaces. In the genus 0 case, our flow is just the harmonic map flow,
and it tries to find branched minimal 2-spheres as in Sacks-Uhlenbeck and
Struwe etc. In the genus 1 case, we show that our flow is exactly equivalent to
that considered by Ding-Li-Lui. In general, we recover the result of Schoen-Yau
and Sacks-Uhlenbeck that an incompressible map from a surface can be adjusted
to a branched minimal immersion with the same action on , and this
minimal immersion will be homotopic to the original map in the case that
.Comment: Updated to reflect galley proof corrections. To appear in The
American Journal of Mathematic
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