Geometry and Topology Seminar
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.