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Geometry and Topology Seminar

Friday, November 8, 2019
3:00pm to 5:00pm
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Linde Hall 187
Comparing complexities of bounded area minimal hypersurfaces
Antoine Song, Department of Mathematics, UC Berkeley,

For a closed minimal surface with area less than A in a Riemannian 3-manifold, there are two natural measures of complexity: its Morse index as a critical point of the area functional, and its genus. How do these two relate? After giving some context, we will prove that they are actually comparable up to a constant factor depending only on the ambient manifold and the area bound A. As we will see, this result generalizes to higher dimensional minimal hypersurfaces with area less than A and with small singular sets in the following way: the index dominates the total Betti number in dimensions 3 to 7, or the size of the singular set in dimensions 8 and above. The proof’s arguments are essentially combinatorial.

For more information, please contact Math Department by phone at 626-395-4335 or by email at mathinfo@caltech.edu.