Asian Journal of Mathematics

Volume 26 (2022)

Number 2

Bounds for the Morse index of free boundary minimal surfaces

Pages: 227 – 252



Vanderson Lima (Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Porto Alegre RS, Brazil)


Inspired by work of Ejiri–Micallef on closed minimal surfaces, we compare the energy index and the area index of a free-boundary minimal surface of a Riemannian manifold with boundary, and show that the area index is controlled from above by the area and the topology of the surface. Combining these results with work of Fraser–Li, we conclude that the area index of a free-boundary minimal surface in a convex domain of Euclidean three-space, is bounded from above by a linear function of its genus and its number of boundary components. We also prove index bounds for submanifolds of higher dimension.


minimal surface, free boundary, index

2010 Mathematics Subject Classification

53A10, 58E12

Received 26 May 2018

Accepted 23 November 2021

Published 6 March 2023