Equivariant index bound for min-max free boundary minimal surfaces

Abstract

Given a three-dimensional Riemannian manifold with boundary and a finite group of orientation-preserving isometries of this manifold, we prove that the equivariant index of a free boundary minimal surface obtained via an equivariant min-max procedure à la Simon-Smith with n-parameters is bounded above by n. Contents 1. Introduction 1 2. Basic notation 5 3. Equivariant spectrum 5 4. Free boundary minimal surfaces with bounded equivariant index 8 5. Outline of the proof of regularity and genus bound 10 6. Deformation theorem 15 7. Proof of the main theorem 18 8. Equivariant index of some families of (free boundary) minimal surfaces 20 Appendix A. Spectrum of elliptic operators with Robin boundary conditions 21 Appendix B. Bumpyness is G-generic 22 References 24

Infinitely many pairs of free boundary minimal surfaces with the same topology and symmetry group

“…There, we collect some results, data and conjectures about the Morse index of the free boundary minimal surfaces we produce. To begin with, we prove in Proposition 7.1 that one can distinguish the surfaces in our family from Ketover's (and thus, conjecturally, from the elements of the Kapouleas-Li family) by their equivariant Morse index as defined in [13]: while the Ketover minimal surfaces have equivariant index equal to one, we prove that the elements in our surfaces have equivariant index at least two. This is a fascinating result, which indicates (among other things) that the surfaces in our family cannot possibly be obtained by means of a one-parameter min-max scheme, but would rather need the use of p-sweepouts for some p ≥ 2 (with numerical evidence indicating that in fact p = 2), modulo the very delicate problem of fully controlling the topology in the procedure.…”

“…The reader is also referred to Section 3 of [13] for a broader discussion of the equivariance constraints.…”

“…Here and throughout the article we let 13) denote the angle between the outward unit conormal to K b and the horizontal plane of equation z = b as shown in Figure 6.…”

“…Firstly, the operator in question has a discrete spectrum, as encoded in the following statement (for its proof see e. g. Appendix A in [13]). Lemma 4.1.…”

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

“…In the following, we recall the definitions of equivariant sweepouts, isotopies, and saturations, as well as the notion of min-max width (cf. [26,23,25] and [7,15]) that are needed for the equivariant min-max construction. Definition 2.1.…”