On the multiplicity one conjecture in min-max theory

DOI10.4007/annals.2020.192.3.3zbMath1461.53051arXiv1901.01173OpenAlexW3098894566WikidataQ122905375 ScholiaQ122905375MaRDI QIDQ2215804

Xin Zhou

Publication date: 14 December 2020

Published in: Annals of Mathematics. Second Series (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1901.01173

zbMATH Keywords

multiplicityminimal hypersurfacesmin-max theoryvolume spectrumhypersurfaces with prescribed mean curvature

Mathematics Subject Classification ID

Minimal surfaces and optimization (49Q05) Existence of solutions for minimax problems (49J35) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42) Variational problems concerning minimal surfaces (problems in two independent variables) (58E12)

Related Items (36)

Generic properties of Steklov eigenfunctions ⋮ Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen-Cahn ⋮ Existence of minimal hypersurfaces with non-empty free Boundary for generic metrics ⋮ Min-max minimal hypersurfaces with obstacle ⋮ Min-max widths of the real projective 3-space ⋮ Min-max theory for free boundary minimal hypersurfaces. II: General Morse index bounds and applications ⋮ On the genus and area of constant mean curvature surfaces with bounded index ⋮ Counting minimal surfaces in negatively curved 3-manifolds ⋮ Existence of constant mean curvature 2‐Spheres in Riemannian 3‐spheres ⋮ An improved Morse index bound of min-max minimal hypersurfaces ⋮ Metrics with \(\lambda_1(-\Delta + k R) \ge 0\) and flexibility in the Riemannian Penrose inequality ⋮ Morse inequalities for the area functional ⋮ Mean curvature and variational theory ⋮ Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere ⋮ Double-well phase transitions are more rigid than minimal hypersurfaces ⋮ A free boundary minimal surface via a 6-sweepout ⋮ The p-widths of a surface ⋮ Existence of infinitely many minimal hypersurfaces in higher-dimensional closed manifolds with generic metrics ⋮ Morse index, Betti numbers, and singular set of bounded area minimal hypersurfaces ⋮ Multiplicity‐1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature ⋮ Existence of multiple closed CMC hypersurfaces with small mean curvature ⋮ Generic uniqueness for the Plateau problem ⋮ Multiplicity one for min-max theory in compact manifolds with boundary and its applications ⋮ Generalized soap bubbles and the topology of manifolds with positive scalar curvature ⋮ Existence of infinitely many free boundary minimal hypersurfaces ⋮ Existence of infinitely many minimal hypersurfaces in closed manifolds ⋮ Generic scarring for minimal hypersurfaces along stable hypersurfaces ⋮ Conformal upper bounds for the volume spectrum ⋮ Min-max theory for free boundary minimal hypersurfaces. I: Regularity theory ⋮ Abundance of minimal surfaces ⋮ Minimal hypersurfaces in the product of two spheres with index one ⋮ Compactness of certain class of singular minimal hypersurfaces ⋮ A comparison of the Almgren-Pitts and the Allen-Cahn min-max theory ⋮ A mountain-pass theorem for asymptotically conical self-expanders ⋮ On the existence of minimal hypersurfaces with arbitrarily large area and Morse index ⋮ Variational aspects of phase transitions with prescribed mean curvature

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