2023/04/30 by Marco A. M. Guaraco, Guaraco, Marco A. M., Stephen Lynch +1 · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2305.00363
Let Γ be a compact codimension-two submanifold of ℝn, and let L be a nontrivial real line bundle over X = ℝn ∖ Γ. We study the Allen--Cahn functional, Eε(u) = ∫X ε (|∇ u|2)/(2) + ((1-|u|2)2)/(4ε) dx, on the space of sections u of L. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to Γ. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as ε → 0, the associated family of energy measures converges to an integer rectifiable (n-1)-varifold V. Moreover, V is stationary with respect to any variation which leaves Γ fixed. Away from Γ, this follows from work of Hutchinson--Tonegawa; our result extends their interior theory up to the boundary Γ. Under additional hypotheses, we can say more about V. When V arises as a limit of critical sections with uniformly bounded Morse index, Σ:= supp ‖V‖ is a minimal hypersurface, smooth away from Γ and a singular set of Hausdorff dimension at most n-8. If the sections are globally energy minimizing and n = 3, then Σ is a smooth surface with boundary, ∂ Σ= Γ (at least if L is chosen correctly), and Σ has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau's problem admits a solution for every boundary curve in ℝ3. This also works if 4 ≤ n≤ 7 and Γ is assumed to lie in a strictly convex hypersurface.