Bounds for the Morse index of free boundary minimal surfaces

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.

Keywords

minimal surface, free boundary, index

2010 Mathematics Subject Classification

53A10, 58E12

Full Text (PDF format)

Received 26 May 2018

Accepted 23 November 2021

Published 6 March 2023