at the

American Institute of Mathematics, San Jose, California

organized by

Lei Ni and Burkhard Wilking

The main topics of the workshop are:

- The Ricci flow and manifolds with positive curvature
The Ricci flow is effective in proving results on the diffeomorphic topological type of manifolds satisfying various positivity assumptions. But there are still many problems that remain open. Interactions between various methods can advance the understandings. A notorious difficult problem in Alexandrov geometry is to smooth singular spaces with lower curvature bound. In particular situations though, one can hope to define a Ricci flow with singular initial data.

- Hypersurface flow, minimal surfaces and eigenvalue estimates
Some effective estimates have been developed for various geometric flow by a `de-regularization' procedure. Typical example is to use the continuity modulus to replace the gradient estimate, expansion modulus to replace the Hessian estimates, Andrews noncollapsing quantity to replace the length of the second fundamental form. These can be used to prove various conjectures involving the eigenvalue estimates and uniquness/rigidity result regarding surfaces with prescribed conditions on the principle curvatures. But there are still problems open.

- Convex geometry and Gauss curvature flow
Fully nonlinear flows on convex surfaces involves both PDE techniques and understandings of convex geometry, which in turn is related to the study of manifolds with positive curvature.

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

A local curvature estimate for the Ricci flow

by Brett Kotschwar, Ovidio Munteanu and Jiaping Wang, *J. Funct. Anal. 271 (2016), no. 9, 2604–2630 * MR3545226

Mean curvature flow of an entire graph evolving away from the heat flow

by Gregory Drugan and Xuan Hien Nguyen, *Proc. Amer. Math. Soc. 145 (2017), no. 2, 861-869 * MR3577885