Differential Geometry Seminar: Ricci flow on manifolds with almost non-negative curvature operator

Seminar | March 6 | 1:10-2 p.m. | 939 Evans Hall

 Richard Bamler, Berkeley

 Department of Mathematics

A celebrated result of Boehm and Wilking states that a Ricci flow starting from a metric whose curvature operator is everywhere positive definite preserves the property of positive curvature operator and converges, modulo rescaling, to a quotient of the round sphere. In contrast, the condition of almost non-negative curvature operator — for example the condition that its smallest eigenvalue is larger than -1 — is not preserved under Ricci flow.

In joint work with Wilking and Cabezas-Rivas we recently showed, however, that a metric whose curvature operator has eigenvalues larger than -1 can be evolved to a Ricci flow on a time-interval of uniform size. Moreover, the eigenvalues of the curvature operator on this flow remain at least bounded from below by a uniform negative constant.

In this talk I will outline the proof of this result and discuss possible applications to the study of manifolds with almost non-negative curvature operator.