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Index estimates of compact hypersurfaces in smooth metric measure spaces

Part of: Classical differential geometry Global differential geometry Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  07 March 2024

Márcio Batista Open the ORCID record for Márcio Batista [Opens in a new window]  and
Matheus B. Martins
Márcio Batista
Affiliation:
CPMAT-IM, Universidade Federal de Alagoas, Maceió, AL 57072-970, Brazil (mhbs@mat.ufal.br; matheus.martins@im.ufal.br)
Matheus B. Martins
Affiliation:
CPMAT-IM, Universidade Federal de Alagoas, Maceió, AL 57072-970, Brazil (mhbs@mat.ufal.br; matheus.martins@im.ufal.br)
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Abstract

In this article, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced an estimate of the Morse index of such hypersurfaces.

Keywords

minimal hypersurfacesMorse indexfree boundaryweighted manifoldsingular manifolds

MSC classification

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Secondary: 58J20: Index theory and related fixed point theorems 53C42: Immersions (minimal, prescribed curvature, tight, etc.)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 5 , October 2025 , pp. 1895 - 1913
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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