NSF Award
1904012
Award: DMS 1711053, Principal Investigator: Yevgeny Liokumovich<br/><br/>A minimal surface is the mathematical idealization of a soap film spanning a wire, which minimizes surface area within the family of spanning surfaces. The min-max theory for minimal surfaces and other variational problems is modeled on a description of an efficient path over a mountain range that goes through a mountain pass: among nearby choices for a road over a mountain, the efficient choice will minimize the maximum altitude attained. A min-max method was developed in the 1960s and 1970s to study existence and other questions for minimal surfaces and has been made more useable in recent years.<br/><br/>These projects address four areas of current min-max theory. An investigation of index and multiplicity bounds is expected to have applications to Heegard surfaces in non-Haken 3-manifolds A second project is intended to develop optimal bounds for min-max families of cycles with integer coefficients and may lead to a related, conjectural parametric coarea inequality. Min-max minimal hypersurfaces in dimensions eight or more may have singularities; a third project will aim to show that for a generic set of metrics on an 8-manifold, smooth minimal hypersurfaces may be constructed. A fourth project concerns equidistributional properties of k-parameter sweepout constructions of minimal hypersurfaces.
