Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-04T13:23:56.663Z Has data issue: false hasContentIssue false
  • English
  • Français

Index estimates of compact hypersurfaces in smooth metric measure spaces

Part of: Global differential geometry Classical 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)
Get access Rights & Permissions [Opens in a new window]

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.)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 19
Check for updates
Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adauto, D. and Batista, M.. Spectrum comparison on free boundary minimal submanifolds of Euclidean domains. Math. Nachrichten 296 (2023), 4673–4685.CrossRefGoogle Scholar
Aiex, N. S.. Index estimate of self-shrinkers in $\Bbb {R}^3$ with asymptotically conical ends. Proc. Am. Math. Soc. 147 (2019), 799–809.CrossRefGoogle Scholar
Aiex, N. S. and Hong, H.. Index estimates for surfaces with constant mean curvature in 3-dimensional manifolds. Calc. Var. Partial Differ. Equ. 60 (2021), 3.CrossRefGoogle Scholar
Ambrozio, L., Carlotto, A. and Sharp, B.. Comparing the Morse index and the first Betti number of minimal hypersurfaces. J. Differ. Geom. 108 (2018a), 379–410.CrossRefGoogle Scholar
Ambrozio, L., Carlotto, A. and Sharp, B.. Index estimates for free boundary minimal hypersurfaces. Math. Ann. 370 (2018b), 1063–1078.CrossRefGoogle Scholar
Bueler, E. L.. The heat kernel weighted Hodge Laplacian on noncompact manifolds. Trans. Am. Math. Soc. 351 (1999), 683–713.CrossRefGoogle Scholar
Castro, K. and Rosales, C.. Free boundary stable hypersurfaces in manifolds with density and rigidity results. J. Geom. Phys. 79 (2014), 14–28.CrossRefGoogle Scholar
Cavalcante, M. P. and de Oliveira, D. F.. Index estimates for free boundary constant mean curvature surfaces. Pac. J. Math. 305 (2020a), 153–163.CrossRefGoogle Scholar
Cavalcante, M. P. and de Oliveira, D. F.. Lower bounds for the index of compact constant mean curvature surfaces in $\Bbb R^3$ and $\Bbb S^3$. Rev. Mat. Iberoam. 36 (2020b), 195–206.CrossRefGoogle Scholar
Hong, H. and Saturnino, A. B.. Capillary surfaces: stability, index and curvature estimates. J. Reine Angew. Math. (Crelles J.) 2023 (2023), 233–265.Google Scholar
Impera, D., Rimoldi, M. and Savo, A.. Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers. Rev. Mat. Iberoam. 36 (2020), 817–840.CrossRefGoogle Scholar
Morgan, F. and Ritoré, M.. Isoperimetric regions in cones. Trans. Am. Math. Soc. 354 (2002), 2327–2339.CrossRefGoogle Scholar
Palmer, B.. Index and stability of harmonic gauss maps. Math. Z. 206 (1991), 563–566.CrossRefGoogle Scholar
Ros, A.. One-sided complete stable minimal surfaces. J. Differ. Geom. 74 (2006), 69–92.CrossRefGoogle Scholar
Sargent, P.. Index bounds for free boundary minimal surfaces of convex bodies. Proc. Am. Math. Soc. 145 (2017), 2467–2480.CrossRefGoogle Scholar
Savo, A.. Index bounds for minimal hypersurfaces of the sphere. Indiana Univ. Math. J. 59 (2010), 823–837.CrossRefGoogle Scholar
Yano, K. (1970) Integral Formulas in Riemannian Geometry. Lecture notes in pure and applied mathematics (Marcel Dekker Inc.).Google Scholar
Zhou, X.. Min-max hypersurface in manifold of positive Ricci curvature. J. Differ. Geom. 105 (2017), 291–343.CrossRefGoogle Scholar
Zhu, J. J.. First stability eigenvalue of singular minimal hypersurfaces in spheres. Calc. Var. Partial Differ. Equ. 57 (2018), 130.CrossRefGoogle Scholar