Geometric Analysis and Topology Seminar

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Volume growth of balls from curvature bounds

Speaker: Gioacchino Antonelli, Notre Dame

Location: Warren Weaver Hall 512

Date: Friday, September 25, 2026, 11 a.m.

Synopsis:

On a complete Riemannian manifold M^n, Ric>=0 and Scal>=1 imply Vol(B_r(p))<=C(n)r^(n-2) for every p in M and r>0. I will present a proof of the latter statement, answering a question posed by Gromov.
 

The proof has two ingredients. First, under the above curvature assumptions, a ball sufficiently close to splitting n-1 Euclidean directions has radius bounded above by a dimensional constant. Second, under nonnegative Ricci curvature alone, a ball sufficiently close to splitting k Euclidean directions, with 0<=k<=n-2, yields another ball close to splitting k+1 directions, while preserving the normalized volume Vol(B_r(p))/r^(n-2) up to a controlled multiplicative constant. Finite iteration gives the desired bound.

Time permitting, I will discuss extensions to positive intermediate curvature and to arbitrary Ricci lower bounds, as well as some consequences. This work was developed with AI assistance.