Volumes of balls in large Riemannian manifolds
Larry Guth
Stanford
May 8, 2007
Abstract: In the 80's,
Gromov made several conjectures about the volumes of balls in
Riemannian manifolds. The spirit of these conjectures is that if
a Riemannian manifold is "large", then it should contain a unit ball
whose volume is not too small. For example, if you take the
standard metric on the n-sphere and increase it pointwise to form a new
metric, then Gromov's conjecture implies that the new metric should
contain a unit ball whose volume is bounded below by a constant
c(n). I proved some of the conjectures, including this one.
I will explain the conjectures and give some context, and then I will
try to say something about the proof.