The principal investigator proposes to study certain elliptic and parabolic problems that arise in differential geometry. One example is the conformal deformation of Riemannian metrics by their scalar curvature (the so-called Yamabe flow). An interesting question is whether the solution approaches a metric of constant scalar curvature or whether the flow may develop a singularity. (The answer is known in dimensions 3 -- 5, but the general case is not yet fully understood.) A solution to this problem will likely require a good understanding of the local properties of the solution in a neighborhood of a blow-up point. Another tool that is likely to be relevant is the positive mass theorem in general relativity. It would also be interesting to see whether similar ideas can shed light on the prescribed scalar curvature problem in higher dimensions.

The proposed project lies at the intersection of three areas of mathematics: differential geometry, nonlinear partial differential equations, and the calculus of variations. One of the goals of this project is to study the deformation of manifolds by their curvature, and to gain a better understanding of the longtime behavior of the solutions. Some of these geometric evolution equation are closely related to equations studied in applied mathematics, such as the porous medium equation.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
0605223
Program Officer
Christopher W. Stark
Project Start
Project End
Budget Start
2006-07-01
Budget End
2009-06-30
Support Year
Fiscal Year
2006
Total Cost
$131,130
Indirect Cost
Name
Stanford University
Department
Type
DUNS #
City
Palo Alto
State
CA
Country
United States
Zip Code
94304