Curvature flows and parabolic PDE on manifolds

Project: Research project

Project Details

Description

Curvature flows and parabolic PDE on manifolds Curvature flows and parabolic PDE on manifolds My recent research has been focused primarily on the Ricci flow and the nature of its singularities in dimensions four and higher. One major project I have been engaged with concerns the classification of noncompact shrinking Ricci solitons (shrinkers) according to their asymptotic type. Growing evidence suggests that every complete noncompact shrinker is either smoothly asymptotic to a cone at infinity or splits locally as a product. As part of a long-term project with L. Wang, I have studied the uniqueness of shrinkers with one of these fixed geometries at infinity. In the first situation, we have shown that two shrinkers which are asymptotic to the same cone at infinity must actually be isometric to each other on a neighborhood of infinity. A refinement of this result shows, in fact, that the isometry group of the cross-section of the cone actually embeds into the isometry group of the end of the shrinker. In the second situation, we have shown so far that two shrinkers which agree to infinite order with one of the standard cylinders on an end must be isometric to those cylinders (or to a quotient thereof). Separately, I have shown that any complete shrinker asymptotic to a Kaehler cone must itself be Kaehler. These works build on techniques from the theory of unique continuation for parabolic and elliptic equations and am I currently working to develop quantitative refinements of the methods used in these works to extend these results and to study the uniqueness of compact singularity models.
StatusActive
Effective start/end date9/1/208/31/25

Funding

  • Simons Foundation: $42,000.00

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