Entropy is a natural geometric quantity measuring the complexity of a surface embedded in R3. 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 R3 which we also establish. To any closed surface in R3 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 R3 with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of non-stationary varifolds (with respect to the Gaussian metric of R3) in a continuous manner.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Geometry and Topology