Morse index of multiplicity one min-max minimal hypersurfaces

By F. C. Marques, A. Neves. Advances in Mathematics, 2015

135 citations recorded by the source corpus.

In this paper, we prove that the Morse index of a multiplicity one, smooth, min-max minimal hypersurface is generically equal to the dimension of the homology class detected by the families used in the construction. This confirms part of the program (\cite{marques-icm}, \cite{marques-neves-cycles}, \cite{marques-neves-index}, \cite{neves-icm}) proposed by the authors with the goal of developing a Morse theory for the area functional.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c160f27b53acb3eeaa374d992231c582f5b11780

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

On the Multiplicity One Conjecture in min-max theory (2019, unknown): “D G ] 5 F eb 2 01 the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one.”
Minimal 2-spheres in 3-spheres (2017, methodology): “t; 2ω1 also fails in two dimensions as the “neck” one would add would contribute signiﬁcant length. 9.1. Multiplicity One Conjecture. Marques-Neves have made the following multiplicity one conjecture [29]: Conjecture 9.7 (Marques-Neves). Given a min-max procedure on a compact manifold, if the metric”
Existence of infinitely many minimal hypersurfaces in closed manifolds (2018, methodology): “The space of relative cycles is endowed with the flat topology and is weakly homotopically equivalent to RP (see [2] and Section 5 of [27] for the case ∂M = ∅).”
Min-max theory for free boundary minimal hypersurfaces, I: Regularity theory (2016, methodology): “ivial closed minimal hypersurface in any given closed Riemannian manifold, and the hypersurface is smooth except possibly on a singular set of codimension 7, (see also other variants and developments [28, 4, 5, 18]). In this paper, we complete Almgren’s program by establishing the regularity theory ”