Index bounds for free boundary minimal surfaces of convex bodies

Abstract

Abstract. In this paper, we give a relationship between the eigenvalues of the Hodge Laplacian and the eigenvalues of the Jacobi operator for a free boundary minimal hypersurface of a Euclidean convex body. We then use this relationship to obtain new index bounds for such minimal hypersurfaces in terms of their topology. In particular, we show that the index of a free boundary minimal surface in a convex domain in R 3 tends to infinity as its genus or the number of boundary components tends to infinity.

Index estimates for free boundary constant mean curvature surfaces

“…We then consider the space of tangential harmonic 1-forms It is well known that these spaces are closed related to the topology of the underline manifold. In fact we have the following result (see [2] or [22]).…”

Index of the critical catenoid

“…They also constructed in [12] free boundary minimal surfaces in B 3 of genus 0 and with an arbitrary number of boundary components. Guided by an hypothetical analogy between (closed) minimal surfaces of S 3 and free boundary minimal surfaces of B 3 , P. Sargent [16] and independently L. Ambrozio, A. Carlotto and B. Sharp [3], have in particular given lower estimates for the Morse index of free boundary minimal surfaces in B 3 , in terms of their topology, similar to the one found by A. Savo [17] for minimal surfaces of the sphere S 3 .…”

“…Sharp [3], have in particular given lower estimates for the Morse index of free boundary minimal surfaces in B 3 , in terms of their topology, similar to the one found by A. Savo [17] for minimal surfaces of the sphere S 3 . Actually, Savo's result also holds in higher dimensions, and the results [16], [3] are much more general than this since they deal with free boundary minimal hypersurfaces in open sets of R n satisfying some convexity assumption (see [3,Theorem 10] for the strongest result so far obtained). Let us also mention the work [2], where Savo's results are in particular extended to closed minimal hypersurfaces inside rank one symmetric spaces.…”

Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

“…Sharp generalized the methods of Ros and Savo and verified the conjecture for a larger class of ambient spaces (see [1]). However, as of today, the full conjecture above is still open, although the method was employed to obtain bounds for minimal free boundary immersions ( [2] and [18]) and complete minimal immersions in R n ( [14]).…”