Morse index and multiplicity of \(\min\text{-}\max\) minimal hypersurfaces. (English) Zbl 1367.49036

The min-max theory for the area functional was started by F. J. Almgren jun. [Topology 1, 257–299 (1962; Zbl 0118.18503)]. Later, in [Existence and regularity of minimal surfaces on Riemannian manifolds. Princeton, New Jersey: Princeton University Press; University of Tokyo Press (1981; Zbl 0462.58003)], J. T. Pitts greatly improved the theory, and the efforts culminated with the proof of the Almgren-Pitts min-max Theorem. The subject may be regarded as a deep higher dimensional generalization of the study of closed geodesics and uses regularity results due to Schoen and Simon. The theory however remained incomplete since it does not provide estimates of the Morse index.
In the paper under review, the authors prove the first general Morse index bounds for minimal hypersurfaces produced by the theory. They also obtain which seems to be the first general multiplicity-one theorem in min-max theory. Motivated by their results, they pose the following:
Multiplicity one conjecture: for generic metrics on \(M^{n+1}\), \(3 \leq (n+1) \leq 7\), two sided instable components of closed minimal hypersurfaces obtained by min-max methods must have multiplicity one.
The paper contains a proof of this conjecture in the case of min-max with one parameter.


49Q20 Variational problems in a geometric measure-theoretic setting
28A75 Length, area, volume, other geometric measure theory
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
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