Macroscopically minimal hypersurfaces

Macroscopically minimal hypersurfaces - Hannah Alpert

Hannah Alpert
Ohio State University
March 12, 2019

A decades-old application of the second variation formula
proves that if the scalar curvature of a closed 3--manifold is bounded
below by that of the product of the hyperbolic plane with the line,
then every 2--sided stable minimal surface has area at least that of
the hyperbolic surface of the same genus. We can prove a coarser
analogue of this statement, taking the appropriate notions of
macroscopic scalar curvature and macroscopic minimizing hypersurface
from Guth's 2010 proof of the systolic inequality for the
n--dimensional torus. The appropriate analogue of hyperbolic area in
this setting turns out to be the Gromov simplicial norm. Joint work
with Kei Funano.