Entropy of closed surfaces and min-max theory. (English) Zbl 1396.53004

Ketover, Daniel; Zhou, Xin

J. Differ. Geom. 110, No. 1, 31-71 (2018).

Summary: Entropy is a natural geometric quantity measuring the complexity of a surface embedded in \(\mathbb{R}^3\). For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured (since proved by Bernstein-Wang) that the entropy of any closed surface is at least that of the self-shrinking two-sphere. In this paper we give an alternative proof of their conjecture for closed embedded 2-spheres. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques-Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in \(\mathbb{R}^3\) which we also establish. To any closed surface in \(\mathbb{R}^3\) we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for \(\mathbb{R}^3\) with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of nonstationary varifolds (with respect to the Gaussian metric of \(\mathbb{R}^3\)) in a continuous manner.

MSC:

53A05

Surfaces in Euclidean and related spaces

Keywords:

complexity of a surface; min-max sequence; self-shrinking plane; varifolds