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A dichotomy for minimal hypersurfaces in manifolds thick at infinity

Song, Antoine (2019) A dichotomy for minimal hypersurfaces in manifolds thick at infinity. . (Unpublished)

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Let (M,g) be a complete (n + 1)-dimensional Riemannian manifold with 2 ≤ n ≤ 6. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose that (M,g) has bounded geometry, or more generally is thick at infinity. Then the following dichotomy holds for the space of closed hypersurfaces in M: either there are infinitely many saddle points of the n-volume functional, or there is none. Additionally, we give a new short proof of the existence of a finite volume minimal hypersurface in finite volume manifolds, we check Yau's conjecture for finite volume hyperbolic 3-manifolds and we extend the density result due to Irie-Marques-Neves when (M,g) is shrinking to zero at infinity.

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Additional Information:The author was partially supported by NSF-DMS-1509027. I am grateful to my advisor Fernando Codá Marques for his crucial guidance. I thank Yevgeny Liokumovich for explaining [2] to me and mentioning [17], [37]. I am thankful to Misha Gromov for exchanges about [17]. I also want to thank Franco Vargas Pallete for discussing with me Yau’s conjecture for finite volume hyperbolic 3-manifolds, a result of which he was also aware. Moreover, a very careful reading by the referees improved the writing of this article.
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Record Number:CaltechAUTHORS:20221026-539140000.8
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117596
Deposited By: George Porter
Deposited On:26 Oct 2022 23:20
Last Modified:26 Oct 2022 23:20

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