Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature
Abstract
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 manifold $N^{n+1}$ with $n\geq 2$ (see Guaraco's 2018 work). We obtain the following multiplicity-$1$ result: if the Ricci curvature of $N$ is positive then the minmax Allen--Cahn solutions concentrate around a multiplicity-$1$ hypersurface, that may have a singular set of dimension $\leq n-7$. This result is new for $n\geq 3$ (for $n=2$ it is also implied by the recent work by Chodosh--Mantoulidis). The argument developed here is geometric in flavour and exploits directly the minmax characterization of the solutions. An immediate corollary is that every compact Riemannian manifold $N^{n+1}$ with $n\geq 2$ and positive Ricci curvature admits a two-sided closed minimal hypersurface, possibly with a singular set of dimension at most $n-7$. This existence result also follows from multiplicity-$1$ results developed within the Almgren--Pitts framework, see works by Ketover-Marques-Neves, Zhou, Marques-Neves, Ramirez-Luna.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.10112
- arXiv:
- arXiv:2004.10112
- Bibcode:
- 2020arXiv200410112B
- Keywords:
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- Mathematics - Analysis of PDEs;
- Mathematics - Differential Geometry;
- 53A10 49Q20 35J20 35J61 35K58 49Q05 53C21 58J60
- E-Print:
- 3 figures, 47 pages. Revised introduction, added references, results unchanged