Min-Max Problems for Families of Cycles in Riemannian Manifolds

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Award: DMS 1711053, Principal Investigator: Yevgeny LiokumovichA 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.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.