• Corpus ID: 203626772

Existence of multiple closed CMC hypersurfaces with small mean curvature

@article{Dey2019ExistenceOM,
  title={Existence of multiple closed CMC hypersurfaces with small mean curvature},
  author={Akashdeep Dey},
  journal={arXiv: Differential Geometry},
  year={2019}
}
  • Akashdeep Dey
  • Published 2 October 2019
  • Mathematics
  • arXiv: Differential Geometry
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 least $\gamma_0c^{-\frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$. This extends the theorem of Zhou and Zhu, where they proved the existence of at least one closed $c$-CMC hypersurface in $(M,g)$. 
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References

SHOWING 1-10 OF 46 REFERENCES
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,
Stable CMC integral varifolds of codimension $1$: regularity and compactness
We give two structural conditions on a codimension $1$ integral $n$-varifold with first variation locally summable to an exponent $p>n$ that imply the following: whenever each orientable portion of
Stable prescribed-mean-curvature integral varifolds of codimension 1: regularity and compactness.
In a previous paper we developed a regularity and compactness theory in Euclidean ambient spaces for codimension 1 weakly stable CMC integral varifolds satisfying two (necessary) structural
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
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
Existence of Infinitely Many Minimal Hypersurfaces in Higher-dimensional Closed Manifolds with Generic Metrics
In this paper, we show that a closed manifold $M^{n+1} (n\geq 7)$ endowed with a $C^\infty$-generic (Baire sense) metric contains infinitely many singular minimal hypersurfaces with optimal
Existence of infinitely many minimal hypersurfaces in positive Ricci curvature
In the early 1980s, S. T. Yau conjectured that any compact Riemannian three-manifold admits an infinite number of closed immersed minimal surfaces. We use min–max theory for the area functional to
Weyl law for the volume spectrum
Given $M$ a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum $\{\omega_p(M)\}_{p\in\mathbb{N}}$ satisfies a Weyl law that was conjectured by Gromov.
Existence of infinitely many minimal hypersurfaces in closed manifolds
Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a
Min–max theory for constant mean curvature hypersurfaces
In this paper, we develop a min–max theory for the construction of constant mean curvature (CMC) hypersurfaces of prescribed mean curvature in an arbitrary closed manifold. As a corollary, we prove
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