Comparing the Morse index and the first Betti number of minimal hypersurfaces

Abstract

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Stability for a Second Type Partitioning Problem

“…We use Ind(M ) to denote the Morse index for a type-II stationary hypersurface M . Following the argument of Savo [25] and Ambrozio-Carlotto-Sharp [2,3], by using the coordinates of harmonic one-forms, we are able to prove the following lower bound for the index. where Rm and Ric denote the Riemannian curvature tensor and Ricci curvature tensor ofM respectively, H ∂B denotes the mean curvature of ∂B ⊂M , and II denotes the second fundamental form for the embeddingM ⊂ R d .…”

“…Since Thanks to above proposition, to estimate the Morse index of M , one only needs to estimate the number of negative eigenvalues of (5.1). Next we use the method of Savo [25], Ambrozio-Carlotto-Sharp [2,3] to find an estimate of number of negative eigenvalues of (5.1) in terms of topological invariant.…”

Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces

“…Compared to the extrinsic flexibility of the method in [ACS18a], Theorem A states that, when the image of the immersion M → R d is contained in a sphere, then the method is actually intrinsically flexible: that is, the linear index bound remains valid under any small deformation of the metric itself.…”

A note on the index of closed minimal hypersurfaces of flat tori

“…We refer the reader to the introduction of [1] for a broader contextualization of this problem and for a discussion of the various cases for which we could verify this conjecture. On the other hand, it is straightforward to observe that inequality (1.1) cannot hold true in the special but fundamental case of flat manifolds, as is seen by considering totally geodesic n-dimensional tori inside (n + 1)-dimensional flat tori (in which case one has index(M) = 0 and b 1 (M) = n for any n ≥ 2).…”

“…In this note we employ the usual musical isomorphisms to pass from vectors to 1-forms, see Remark 2.1 in [1] for further details.…”