Parabolic Differential Equations and the Geometry of Manifolds

Project: Research project

Description

Thus far, the principal achievements of the research sponsored by this grant have been several unique-continuation-type results for a class of weakly-parabolic and elliptic PDE with geometric degeneracies, including the Ricci and meancurvature flows and the Ricci-soliton equation. Among these results is a proof of the backwards-uniqueness of solutions to the Ricci flow, a consequence of which is that the isometry group of a solution cannot expand during the lifetime of the solution; the results in this paper have since been published in IMRN. With an extension of this technique (detailed in a recently submitted manuscript), I have been able to establish the non-contraction of the holonomy group of a solution to Ricci flow this is a backwards-time analog of a result of Hamilton, and together the results imply that the holonomy group of a solution is invariant under the flow. I have also verified a similar result for solutions to mean curvature flow; the details are in a paper under current preparation. In another paper under preparation, using an approach of W. Wong and P. Yu, I have been able to establish a local unique-continuation property of the Ricci soliton equation which avoids an appeal to analyticity in special coordinates. Concerning the other areas outlined in my proposal, in a preprint, I describe of a geometric realization of the full Harnack quantity for curvature flows of convex hypersurfaces and use this framework to give a new proof of Hamiltons inequality for the mean curvature flow. I also have some as-yet-unpublished findings regarding the connection between this construction and the epsilonregularization procedure of Evans-Spruck in the level-set formulation of the mean curvature flow. Finally, I have also devoted some effort to the study of certain rigidity conjectures for gradient Ricci solitons, and have obtained some partial results, however, the work in this component of my proposal remains very much an ongoing project. Throughout the project thus far, and with the help of the awards sponsorship, I have also presented these results to my colleagues and a broader community of mathematicians as participant and speaker in a variety of workshops, meetings, seminars, conferences and colloquia, in the U.S. and abroad.
StatusFinished
Effective start/end date12/8/118/31/13

Funding

  • National Science Foundation (NSF): $25,937.00

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Parabolic Differential Equations
Ricci Soliton
Mean Curvature Flow
Holonomy Group
Unique Continuation
Soliton Equation
Ricci Flow
Preparation
Curvature Flow
Parabolic PDEs
Elliptic PDE
Isometry Group
Uniqueness of Solutions
Appeal
Analyticity
Level Set
Rigidity
Expand
Hypersurface
Lifetime