Allen-Cahn minmax and multiplicity-1 minimal hypersurfaces

Costante Bellettini, University College London

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The existence of a closed minimal hypersurface in a compact Riemannian manifold was first established by the combined efforts of Almgren, Pitts, Schoen-Simon-Yau, Schoen-Simon in the early 80s by means of what is nowadays called Almgren-Pitts minmax. An alternative approach to reach the same existence result has been implemented in recent years in a work by Guaraco, using a minmax construction for the Allen-Cahn energy, in combination with works by Hutchinson-Tonegawa, Tonegawa, Tonegawa-Wickramasekera, Wickramasekera. A natural question (ubiquitous in geometric analysis and, in particular, in minmax constructions) is whether the minimal hypersurface is obtained with multiplicity 1. The multiplicity-1 information has important geometric consequences; however, the a priori possibility of higher multiplicity is intrinsic in both minmax constructions, as they are carried out in the class of varifolds. After an overview, this talk focuses on the case of an ambient Riemannian manifold (of dimension 3 or higher) with positive Ricci curvature: in this case, the minmax construction via Allen-Cahn yields a multiplicity-1 minimal hypersurface.