• Corpus ID: 220845778

# An Improved Morse Index Bound of Min-Max Minimal Hypersurfaces.

@article{Li2020AnIM,
title={An Improved Morse Index Bound of Min-Max Minimal Hypersurfaces.},
author={Yangyang Li},
journal={arXiv: Differential Geometry},
year={2020}
}
• Yangyang Li
• Published 28 July 2020
• Mathematics
• arXiv: Differential Geometry
In this paper, we give an improved Morse index bound of minimal hypersurfaces from Almgren-Pitts min-max construction in any closed Riemannian manifold $M^{n+1}$ $(n+1 \geq 3$), which generalizes a result by X. Zhou \cite{zhou_multiplicity_2019} for $3 \leq n+1 \leq 7$. The novel techniques are the construction of hierarchical deformations and the restrictive min-max theory. These techniques do not rely on bumpy metrics, and thus could be adapted to many other min-max settings.

## Figures from this paper

Minimal hypersurfaces for generic metrics in dimension 8
• Mathematics
• 2022
. We show that in an 8-dimensional closed Riemmanian manifold with C ∞ -generic metrics, every minimal hypersurface is smooth and nondegenerate. This conﬁrms a full generic regularity conjecture of
On the Genus and Area of Constant Mean Curvature Surfaces with Bounded Index
Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh, Ketover and Maximo we show that in any closed Riemannian 3-manifold $(M,g)$, the genus of an
SOME NEW GENERIC REGULARITY RESULTS FOR MINIMAL SURFACES AND MEAN CURVATURE FLOWS LECTURE NOTES FOR GEOMETRIC ANALYSIS FESTIVAL, 2021
For Γn−1 ⊂ ∂B1 ⊂ R, consider Σ ⊂ B1 a hypersurface with ∂Σ = Γ, with least area among all such surface. (This is known as the Plateau problem). It might happen that Σ is singular. For example,
Morse inequalities for the area functional
• Mathematics
• 2020
In this article we prove the strong Morse inequalities for the area functional in codimension one, assuming that the ambient dimension satisfies $3 \leq (n + 1) \leq 7$, in both the closed and the

## References

SHOWING 1-10 OF 34 REFERENCES
Morse index of multiplicity one min-max minimal hypersurfaces
• Mathematics
• 2021
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
Orientability of min-max hypersurfaces in manifolds of positive Ricci curvature
Let $M^{n+1}$ be an orientable compact Riemannian manifold with positive Ricci curvature. We prove that the Almgren-Pitts width of $M^{n+1}$ is achieved by an orientable index $1$ minimal
Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature
We address the one-parameter minmax construction, via Allen--Cahn energy, that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian
On the Multiplicity One Conjecture in min-max theory
• Xin Zhou
• Mathematics
Annals of Mathematics
• 2020
We prove that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves,
Density of minimal hypersurfaces for generic metrics.
• Mathematics
• 2017
For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is
Compactness of certain class of singular minimal hypersurfaces
• Akashdeep Dey
• Mathematics
Calculus of Variations and Partial Differential Equations
• 2021
Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the
Equidistribution of minimal hypersurfaces for generic metrics
• Mathematics
Inventiones mathematicae
• 2019
AbstractFor almost all Riemannian metrics (in the $$C^\infty$$C∞ Baire sense) on a closed manifold $$M^{n+1}$$Mn+1, $$3\le (n+1)\le 7$$3≤(n+1)≤7, we prove that there is a sequence of closed, smooth,
Existence of hypersurfaces with prescribed mean curvature I – generic min-max
• Mathematics
Cambridge Journal of Mathematics
• 2020
We prove that, for a generic set of smooth prescription functions $h$ on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature $h$. The
A dichotomy for minimal hypersurfaces in manifolds thick at infinity
Let $(M^{n+1},g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of Yau's conjecture for minimal surfaces and builds on a result
Existence of multiple closed CMC hypersurfaces with small mean curvature
Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0 0$, there exist at