Morse index and multiplicity of min-max minimal hypersurfaces

Abstract

The Min-max Theory for the area functional, started by Almgren in the early 1960s and greatly improved by Pitts in 1981, was left incomplete because it gave no Morse index estimate for the minmax minimal hypersurface.We advance the theory further and prove the first general Morse index bounds for minimal hypersurfaces produced by it. We also settle the multiplicity problem for the classical case of one-parameter sweepouts.

Morse theory for the area functional

“…Combining the results of [14], [16] and [35], for a generic metric g, there exists a closed minimal hypersurface Σ k ⊂ (M n+1 , g) with index(Σ k ) = k for every k ∈ N. Since b k (Z n (M n+1 , Z 2 )) = 1, this corresponds to the inequality c k ≥ b k . Notice that the area of Σ k is equal to the k-width ω k (M, g) of M (see [16] for the definition). For more details see the survey paper [17].…”

“…Notice that in Section 3 of [14], a homotopy class is defined in an unusual way: the homotopy class of an F-continuous map is defined as the class of all F-continuous maps that are homotopic to the original one in the flat topology. But Proposition 3.5 of [15] has been upgraded to the mass topology in Proposition 3.2 of [16]. An inspection of Section 3 of [14] shows that the min-max theory also works for usual F-continuous homotopy classes.…”

“…But Proposition 3.5 of [15] has been upgraded to the mass topology in Proposition 3.2 of [16]. An inspection of Section 3 of [14] shows that the min-max theory also works for usual F-continuous homotopy classes. Deformation Theorem A of [14] also applies to to rule out minimal hypersurfaces of higher index.…”

“…Since W (σ) ≥ s, we have that the techniques of [16] for standard homology classes can be adapted to the case of relative homology classes as in the statement of the theorem. There is δ > 0 such that for sufficiently large j,…”

“…The point was to show that generically, in which case the metric is bumpy meaning closed minimal hypersurfaces are nondegenerate (White [30], [34]), a closed minimal hypersurface produced by min-max over kparameter sweepouts must have index equal to k. General upper bounds were obtained in [14] and lower bounds were proven in [16] assuming the Multiplicity One Conjecture: any component of a min-max minimal hypersurface has multiplicity one. As in [16], our paper uses the local min-max property of White [31].…”