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Index estimates for closed minimal submanifolds of the sphere.
- Source :
- Proceedings of the Royal Society of Edinburgh: Section A: Mathematics; Jun2022, Vol. 152 Issue 3, p802-816, 15p
- Publication Year :
- 2022
-
Abstract
- In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$ -forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03082105
- Volume :
- 152
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- Proceedings of the Royal Society of Edinburgh: Section A: Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 157192771
- Full Text :
- https://doi.org/10.1017/prm.2021.86