Macroscopically minimal hypersurfaces

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.

Date

Speakers

Hannah Alpert

Affiliation

Ohio State University