• 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. 
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References

SHOWING 1-10 OF 34 REFERENCES
Morse index of multiplicity one min-max minimal hypersurfaces
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.
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
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
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…
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