Extend the foliated Plateau problem to higher dimensions

Extend the foliated Plateau problem from three-dimensional Riemannian balls, where minimal surfaces or elliptic-curvature discs foliate the unit tangent bundle, to higher-dimensional manifolds by constructing analogous foliations by minimal hypersurfaces or appropriate curvature hypersurfaces and developing a higher-dimensional replacement for the graph-theoretic analysis of Jacobi-equation zero sets.

Background

The paper’s geometric and microlocal arguments depend essentially on solving a foliated Plateau problem in dimension three. In that setting, the Gauss lifts of discs spanning a three-dimensional family of boundary circles foliate the unit tangent bundle, and the two-dimensional topology of the discs permits a graph-theoretic analysis of Jacobi fields’ zero sets.

For higher-dimensional analogues, the discs would be replaced by higher-dimensional balls and the boundary circles by hyperspheres. The authors identify the absence of a satisfactory higher-dimensional analogue of their zero-set argument as the principal obstruction.