Free boundary minimal surfaces with connected boundary and arbitrary genus

Carlotto, Alessandro; Franz, Giada; Schulz, Mario B.

doi:10.48550/arxiv.2001.04920

Cited by 10 publications

(20 citation statements)

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“…Applications. As a direct application, we deduce that the family of free boundary minimal surfaces in B 3 constructed in [CFS20] have dihedral equivariant index equal to one.…”

Section: Introductionmentioning

confidence: 86%

“…One of its primary goals is to construct new families of minimal surfaces in S 3 and of free boundary minimal surfaces in B 3 . Indeed, encoding the right symmetry group in the min-max procedure allows us to produce surfaces with fully controlled topology, as done by the author in joint work with Carlotto and Schulz in [CFS20].…”

Section: Introductionmentioning

confidence: 99%

“…Let us briefly recall the geometry of these surfaces in order to better appreciate the theorem below. Fixed any g ≥ 1, recall that the dihedral group D g+1 acting on B 3 is defined as the subgroup of Euclidean isometries generated by the rotation of angle 2π/(g + 1) around the vertical axis ξ 0 := {(0, 0, r) : r ∈ [−1, 1]} and by the rotations of angle π around the g + 1 horizontal axes ξ k := {(r cos(kπ/(g + 1)), r sin(kπ/(g + 1)), 0) : r ∈ [−1, 1]} (see also [CFS20,pp. 1]).…”

Section: Introductionmentioning

confidence: 99%

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Equivariant index bound for min-max free boundary minimal surfaces

Franz¹

2021

Preprint

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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

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“…Applications. As a direct application, we deduce that the family of free boundary minimal surfaces in B 3 constructed in [CFS20] have dihedral equivariant index equal to one.…”

Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivariant index bound for min-max free boundary minimal surfaces

Franz¹

2021

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“…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.…”

Section: Construction Of the Sweepoutmentioning

confidence: 99%

“…In general, the convergence of a min-max sequence is only obtained in the sense of varifolds and additional work is required to control the topology of the limit surface. Our argument to determine the genus differs from the approach described in [24] and in particular relies on our Lemma 2.9 about the structure of arbitrary closed, equivariant surfaces, allowing a generalisation of [7,Lemma B.1] used in recent work of the third author in collaboration with Carlotto and Franz. Moreover, proving the behaviour for high genus as stated in Theorem 1.3 is rather delicate and we rely again on a careful application of Lemma 2.9.…”

Section: Introductionmentioning

confidence: 99%

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Buzano¹,

Nguyen²,

Schulz³

2021

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In his lecture notes on mean curvature flow, Ilmanen conjectured the existence of noncompact self-shrinkers with arbitrary genus. Here, we employ min-max techniques to give a rigorous existence proof for these surfaces. Conjecturally, the self-shrinkers that we obtain have precisely one (asymptotically conical) end. We confirm this for large genus via a precise analysis of the limiting object of sequences of such self-shrinkers for which the genus tends to infinity. Finally, we provide numerical evidence for a further family of noncompact self-shrinkers with odd genus and two asymptotically conical ends.

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Shape optimization for combinations of Steklov eigenvalues on Riemannian surfaces

Petrides

2024

Math. Z.

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We prove existence and regularity of metrics which minimize combinations of Steklov eigenvalues over metrics of unit perimeter on a surface with boundary. We show that there are free boundary minimal immersions into ellipsoids parametrized by eigenvalues, such that the coordinate functions are eigenfunctions with respect to the minimal metrics. This work generalizes Fraser-Schoen's and the author's maximization for one eigenvalue among metrics of unit perimeter on a surface giving free boundary minimal immersions into balls. We also generalize the previous results of critical metrics for one eigenvalue to any combination of eigenvalues from target balls to target ellipsoids.

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Free boundary minimal surfaces with connected boundary and arbitrary genus

Cited by 10 publications

References 0 publications

Equivariant index bound for min-max free boundary minimal surfaces

Equivariant index bound for min-max free boundary minimal surfaces

Noncompact self-shrinkers for mean curvature flow with arbitrary genus

Shape optimization for combinations of Steklov eigenvalues on Riemannian surfaces

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