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Index estimates for closed minimal submanifolds of the sphere

Published online by Cambridge University Press:  17 December 2021

Diego Adauto
Affiliation:
DME, Universidade do Estado do Rio Grande do Norte, Mossoró, RN, 59610-210, Brazil (diegoalves@uern.br)
Márcio Batista
Affiliation:
CPMAT-IM, Universidade Federal de Alagoas, Maceió, AL, 57072-970, Brazil (mhbs@mat.ufal.br)

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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