Index and first Betti number of f-minimal hypersurfaces and\n self-shrinkers

2018/03/22 by Debora Impera, Impera, Debora, Michele Rimoldi +3 · 3 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1803.08268

openalex publication_date 2018/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Morse index of self-shrinkers for the mean curvature flow and,\nmore generally, of f-minimal hypersurfaces in a weighted Euclidean space\nendowed with a convex weight. When the hypersurface is compact, we show that\nthe index is bounded from below by an affine function of its first Betti\nnumber. When the first Betti number is large, this improves index estimates\nknown in literature. In the complete non-compact case, the lower bound is in\nterms of the dimension of the space of weighted square summable f-harmonic\n1-forms; in particular, in dimension 2, the procedure gives an index\nestimate in terms of the genus of the surface.\n

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• A weighted Reilly type integral formula for differential forms and its applications
• The first nonzero eigenvalue of the weighted p-Laplacian on differential forms
• Stability of weighted minimal hypersurfaces under a lower 1-weighted Ricci curvature bound

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