# Geometric Analysis and Topology Seminar

#### A generalization of Geroch's conjecture

**Speaker:**
Florian Johne, Columbia University

**Location:**
Warren Weaver Hall 512

**Date:**
Friday, October 28, 2022, 11 a.m.

**Synopsis:**

Closed manifolds with topology N = M x S^1 do not admit metrics of positive Ricci curvature by the theorem of Bonnet-Myers, while the the resolution of the Geroch conjecture implies that the torus T^n does not admit a metric of positive scalar curvature. In this talk we explain a non-existence result for metrics of positive m-intermediate curvature (a notion of curvature reducing to Ricci curvature for m = 1, and scalar curvature for m = n-1) on closed manifolds with topology N^n = M^{n-m} x T^m for n <= 7. Our proof uses minimization of weighted areas, the associated stability inequality, and delicate estimates on the second fundamental form. This is joint work with Simon Brendle and Sven Hirsch.