Min-max theory in Gaussian space and Entropy Conjecture

Min-max theory in Gaussian space and Entropy Conjecture

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Xin Zhou , MIT
Fine Hall 314

Please note additional talk and special time (4:15).   Minimal surfaces are critical points of the area functional. The min-max theory is a variational theory for constructing saddle point type, unstable minimal surfaces. In this talk, we will introduce a min-max theory in a specific space--the Gaussian probability space. Minimal surfaces in Gaussian space are also called self-shrinkers, which model the singularities of the Mean Curvature Flow. Self-shrinkers are unstable with respect to the second variation of area. Any variational construction of self-shrinkers must be of min-max type.  As an application, we will address a conjecture concerning the entropy of closed surfaces. The entropy is a quantity which measures complexity of a surface. Colding-Ilmanen-Minicozzi-White conjectured that the entropy of a closed surface is bounded from below by that of a round sphere. We will give a min-max proof of this conjecture for spheres. In fact, the entropy is in many ways similar to that of the Willmore functional, and our argument is analogous in many ways to that of Marques-Neves on the Willmore problem. This is based on a joint work with Dan Ketover.