PIMS Differential Geometry and Analysis Summer School
August 13 - August 17, 2007
University of Washington
Department of Mathematics
Seattle, WA 98195
|
PIMS Differential Geometry and Analysis Summer School August 13 - August 17, 2007 |
|
|
University of Washington |
Richard M. Schoen
Stanford University
"Riemannian Manifolds of Positive Curvature"
In these lectures we will describe geometric PDE methods of studying manifolds of positive curvature for a
variety of notions of positivity. The two basic methods on which we will focus will be the theory of
minimal submanifolds and second variation of volume, and the Ricci flow. We will discuss the PIC condition
introduced by Micallef and Moore, and present our recent proof (joint with S. Brendle) of the
differentiable sphere theorem for manifolds with positive 1/4-pinched curvature.
The first papers to read are:
(1) M. Micallef and J. D. Moore, Annals of Math 127, 199-227
(1988)
(2) R. Hamilton, JDG 24, 153-179 (1986)
Next you should read:
(1) A. Fraser, Annals of Math. 158, 345-354 (2003)
(2) A. Fraser and J. Wolfson, Duke Math J. 133, 325-334 (2006)
(3) C. Böhm and B. Wilking, "Manifolds
with positive curvature operator are space forms," to appear in Annals of Math
(4) S. Brendle and R. Schoen, "Manifolds
with 1/4-pinched curvature are space forms"