We show that the Morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the Euclidean space grows linearly with the dimension of its first relative homology group (which is at least as big as the number of its boundary components, minus one). In ambient dimension three, this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components. Thereby, the compactness theorem by Fraser and Li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded Morse index inside a convex domain. Our estimates also imply that the examples constructed, in the unit ball, by Fraser–Schoen and Folha–Pacard–Zolotareva have arbitrarily large index. Extensions of our results to more general settings (including various classes of positively curved Riemannian manifolds and other convexity assumptions) are discussed.
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The authors would like to thank Ivaldo Nunes, Fernando Codá Marques and André Neves for their interest in this work. L. A. is supported by the ERC Start Grant PSC and LMCF 278940 and would like to thank the Scuola Normale Superiore where part of this project was completed. This article was done while A. C. was an ETH-ITS fellow: the outstanding support of Dr. Max Rössler, of the Walter Haefner Foundation and of the ETH Zürich Foundation are gratefully acknowledged. B.S. would like to thank the ETH-FIM for their hospitality and excellent working environment during the completion of this project. B.S. was partially supported by the Scuola Normale Superiore (Commissione Ricerca, Progetto Giovani Ricercatori).
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Ambrozio, L., Carlotto, A. & Sharp, B. Index estimates for free boundary minimal hypersurfaces. Math. Ann. 370, 1063–1078 (2018). https://doi.org/10.1007/s00208-017-1549-8