Award: DMS 1406337, Principal Investigator: Xin Zhou
The physical description of a soap film spanning a wire boundary leads to an important class of surfaces in geometry, called "minimal surface", because the energy-minimization condition that the soap film satisfies corresponds to a surface of least area, at least among nearby surfaces. Also as intrinsic geometric objects of the underlying space, minimal surfaces can be viewed as non-linear analog of the eigenstates in Quantum Mechanics. The mathematical problem that amounts to showing that every bounding wire is spanned by at least one soap film has been studied in detail for many years and admits many important generalizations. Most of the previous techniques for proving such existence properties focused on the "minimizing theory", which corresponds to finding the ground state in Quantum Mechanics. One natural but very difficult technique, called the "min-max theory", corresponding to finding the excited states in Quantum Mechanics (i.e. the eigenstates with energy higher than the ground state), has had striking recent successes, and will be one of the major objects of study in this research program. Another project will study geometric inequalities in general relativity, particularly aiming to understand the mass angular momentum inequalities for axisymmetric initial data sets which model rotating galaxies in astrophysics.
More specifically, the principal investigator will study the min-max theory for minimal surfaces via both the geometric measure theory approach and the harmonic map approach. Concerning the geometric measure theory approach, some of the aspects of the min-max theory to be studied in this research program include extensions of the PI's index bound theorem beyond positive Ricci curvature conditions, as well as versions of the Almgren-Pitts min-max theory for more general ambient spaces, including manifolds with boundary and smooth metric measure spaces (where the Bakry-Emery Ricci tensor plays a role). Certain geometric problems related to the Min-max theory will be studied, such as Hersch-type estimates for higher-dimensional minimal hypersurfaces with bounded Morse index. Concerning the min-max theory via harmonic maps, the PI intends to study the issue of Morse index and branch points of the min-max minimal surfaces constructed by Colding-Minicozzi and himself, and the corresponding free boundary problem.
