Minimal Surfaces in Geometry and Topology

Minimal surfaces are shapes in equilibria first studied by Lagrange in the 1700s. Such surfaces locally minimize area and thus are in a sense optimal and ubiquitous - in chemistry, materials science, biology and general relativity (where they model apparent horizons of black holes). In mathematics, they have been used more recently to solve problems in Geometry and Topology, such as in the proof of the Poincare conjecture. The PI will study the construction, properties and applications of minimal surfaces in three-dimensional spaces. A central problem is to understand the geometry and topological type of the minimal surfaces one can obtain. In topology, a question asked by J.W. Alexander in 1932 is to find all the ways to divide a three-dimensional space into two pieces of a simpler type. Minimal surfaces can be used as canonical surfaces to find such splittings. The PI will also study related problems arising from variational principles, for instance the problem of Arnold which asks to find closed orbits of a particle subject to a magnetic field. In addition to this research, the PI will focus on teaching and training of undergraduate and graduate students as well as advancing the field by organizing seminars, conferences and writing expository materials. More precisely, the objectives of this project are to develop new techniques to study minimal surfaces arising from the smooth min-max theory of Simon-Smith. When minimal surfaces are constructed from multi-parameter sweep-outs, a basic and important open problem is whether they come with integer multiplicities. The PI will work to show that the multiplicities are generically equal to one in the smooth setting. One goal in this direction is to prove the Lusternick-Schnirelman Conjecture that every Riemannian three-sphere contains at least four embedded minimal two-spheres. Such work requires developing quantitative versions of topological theorems, such as Cerf's theorem. The PI will also use minimal surfaces to continue the study of classifying Heegaard splittings of three-manifolds. The PI will also investigate related variational problems involving mean curvature such as obtaining the existence of closed curves of constant curvature on Riemannian two-spheres, a problem originating from physics and dynamical systems. The techniques to be employed in these projects combine ideas from low-dimensional topology, analysis, Morse theory, and minimal surface theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.