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} }
In this paper, we show that every $8$-dimensional closed Riemmanian manifold with $C^\infty$-generic metrics admits a smooth minimal hypersurface.
5 Citations
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…
Mean Convex Smoothing of Mean Convex Cones
- Mathematics
- 2022
We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a…
Singular behavior and generic regularity of min-max minimal hypersurfaces
- Mathematics
- 2020
We show that for a generic $8$-dimensional Riemannian manifold with positive Ricci curvature, there exists a smooth minimal hypersurface. Without the curvature condition, we show that for a dense set…
SOME NEW GENERIC REGULARITY RESULTS FOR MINIMAL SURFACES AND MEAN CURVATURE FLOWS LECTURE NOTES FOR GEOMETRIC ANALYSIS FESTIVAL, 2021
- Mathematics
- 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,…
Mean curvature flow with generic low-entropy initial data
- Mathematics
- 2021
. We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their mean curvature flow encounters only spherical and cylindrical singularities. Our theorem applies to all…
References
SHOWING 1-10 OF 62 REFERENCES
Existence of Infinitely Many Minimal Hypersurfaces in Higher-dimensional Closed Manifolds with Generic Metrics
- Mathematics
- 2019
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…
On the bumpy metrics theorem for minimal submanifolds
- Mathematics
- 2015
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…
Compactness of minimal hypersurfaces with bounded index
- Mathematics
- 2015
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…
Existence of minimal hypersurfaces in complete manifolds of finite volume
- MathematicsInventiones 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…
Min–max theory for constant mean curvature hypersurfaces
- MathematicsInventiones 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…
Singular homologically area minimizing surfaces of codimension one in Riemannian manifolds
- Mathematics
- 1999
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…
Approximation by smooth embedded hypersurfaces with positive mean curvature
- MathematicsBulletin 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…
Existence of hypersurfaces with prescribed mean curvature I – generic min-max
- MathematicsCambridge 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…
On the Multiplicity One Conjecture in min-max theory
- MathematicsAnnals 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,…
A general regularity theory for stable codimension 1 integral varifolds
- Mathematics
- 2009
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…