• Corpus ID: 228083896

# Generic Regularity of Minimal Hypersurfaces in Dimension 8.

@article{Li2020GenericRO,
title={Generic Regularity of Minimal Hypersurfaces in Dimension 8.},
author={Yangyang Li and Zhihan Wang},
journal={arXiv: Differential Geometry},
year={2020}
}
• Published 10 December 2020
• Mathematics
• arXiv: Differential Geometry
In this paper, we show that every $8$-dimensional closed Riemmanian manifold with $C^\infty$-generic metrics admits a smooth minimal hypersurface.
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## References

SHOWING 1-10 OF 62 REFERENCES

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
This paper proves several natural generalizations of the theorem that for a generic, $C^k$ Riemannian metric on a smooth manifold, there are no closed, embedded, minimal submanifolds with nontrivial
We prove a compactness result for minimal hypersurfaces with bounded index and volume, which can be thought of as an extension of the compactness theorem of Choi-Schoen (Invent. Math. 1985) to higher
• Mathematics
Inventiones mathematicae
• 2019
We prove that every complete non-compact manifold of finite volume contains a (possibly non-compact) minimal hypersurface of finite volume. The main tool is the following result of independent
• Mathematics
Inventiones mathematicae
• 2019
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
Abstract. For n≥7, it is shown how to construct examples of smooth, compact Riemannian manifolds (Nn+1,g), with non-trivial n dimensional integer homology, such that for some Γ∈Hn(N,Z), the
• F. Lin
• Mathematics
Bulletin of the Australian Mathematical Society
• 1987
Here we initiate the study of the following problem. Let Ω be a compact domain in a Riemannian manifold such that ∂Ω is of minimum area for the contained volume. Can ∂Ω be approximated by smooth
• 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
• 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,
We give a necessary and sufficient geometric structural condition, which we call the a -Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to