Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory
By M. Li, Xin Zhou. Journal of differential geometry, 2016
87 citations recorded by the source corpus.
In 1960s, Almgren initiated a program to find minimal hypersurfaces in compact manifolds using min-max method. This program was largely advanced by Pitts and Schoen-Simon in 1980s when the manifold has no boundary. In this paper, we finish this program for general compact manifold with nonempty boundary. As a result, we prove the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. An application of our general existence result combined with the work of Marques and Neves shows that for any compact Riemannian manifolds with nonnegative Ricci curvature and convex boundary, there exist infinitely many embedded minimal hypersurfaces with free boundary which are properly embedded.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/054a6794501cd981030b4418e3239df2bd61b297
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Min–max theory for constant mean curvature hypersurfaces (2017, background): “Due to these tremendous successes, there have been a vast number of developments of this program in various contexts, including [9,13,14,18,25,26,28,29,34,39,54].”
- Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “Zhou [21] (their results hold for more general sweepouts as defined in [26, 22]).”
- On the Multiplicity One Conjecture in min-max theory (2019, unknown): “We refer to [39, 6, 10, 17, 9, 23, 7, 47, 50, 32] for other variants of this theory.”
- Existence of hypersurfaces with prescribed mean curvature I – generic min-max (2018, background): “Consequently, the min-max program has seen a series of developments in various directions, including (but not limited to) [16, 28, 37, 44, 42, 15, 41, 36, 12, 46, 57].”