Extremal Eigenvalue Problems and Free Boundary Minimal Surfaces in the Ball

Fraser, Ailana

doi:10.1007/978-3-030-53725-8_1

Cited by 4 publications

(3 citation statements)

References 40 publications

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“…In the last decade, we have witnessed a striking development of the theory of free boundary minimal surfaces, to an extent that cannot be properly accounted for here. We refer the reader to the recent survey [41], to the lecture notes [15] and to the introduction of the PhD thesis [14] -among others -for an overview of some of the most significant advances in the field. Nevertheless, one cannot but note the abundance of open problems, including a few that have proven to be very elusive.…”

Section: Introductionmentioning

confidence: 99%

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

Carlotto¹,

Schulz²,

Wiygul³

2022

Preprint

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The topology and symmetry group of a free boundary minimal surface in the three-dimensional Euclidean unit ball do not determine the surface uniquely. We provide pairs of non-isometric free boundary minimal surfaces having any sufficiently large genus g, three boundary components and antiprismatic symmetry group of order 4(g + 1). A. Parametrization of theKarcher-Scherk towers B. Analysis on asymptotically cylindrical surfaces C. Graphical deformations of immersed hypersurfaces D. Asymptotic behavior of the Ketover free boundary minimal surfaces 1

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Section: Introductionmentioning

confidence: 99%

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

Carlotto¹,

Schulz²,

Wiygul³

2022

Preprint

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“…Recently, thanks to the breakthroughs in the Almgren-Pitts min-max theory, there has been substantial progress in the general existence and regularity theory for free boundary minimal hypersurfaces in Riemannian manifolds. Please consult [5,14,20,27], and so on, for recent progresses in free boundary minimal surface theory.…”

Section: Introductionmentioning

confidence: 99%

Rigidity and vanishing theorems for submanifolds with free boundary in the unit ball

Zhao,

Cao

2024

Bulletin of London Math Soc

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In this paper, we investigate the rigidity and vanishing properties of compact submanifolds with free boundary of arbitrary codimension in the unit ball. We first show that a minimal submanifold with free boundary in the unit ball satisfying a pointwise or integral curvature pinching condition on the second fundamental form is a flat equatorial disk. Then we prove a vanishing theorem for cohomology groups for submanifolds with free boundary in the unit ball under an integral curvature pinching condition on the trace‐free second fundamental form.

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“…Despite a profusion of constructions of free boundary minimal surfaces in the Euclidean unit ball B 3 over the course of the past decade ( [14-16, 24, 31] via optimization of the first Steklov eigenvalue, [4,25,26] via min-max methods for the area functional, and [6,11,[18][19][20]22] via gluing methods), many basic questions about the space of such surfaces remain open. The reader is referred to [12,13,27] for recent overviews of the field. In particular, so far it is only for the rotationally symmetric examples, planar discs through the origin and critical catenoids, that the exact value of the Morse index is actually known (see [8,36,38]).…”

Section: Introductionmentioning

confidence: 99%

Spectral estimates for free boundary minimal surfaces via Montiel-Ros partitioning methods

Carlotto¹,

Schulz²,

Wiygul³

2023

Preprint

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We adapt and extend the Montiel-Ros methodology to compact manifolds with boundary, allowing for mixed (including oblique) boundary conditions and also accounting for the action of a finite group G together with an additional twisting homomorphism σ : G → O(1). We then apply this machinery in order to obtain quantitative lower and upper bounds on the growth rate of the Morse index of free boundary minimal surfaces with respect to the topological data (i. e. the genus and the number of boundary components) of the surfaces in question. In particular, we compute the exact values of the equivariant Morse index and nullity for two infinite families of examples, with respect to their maximal symmetry groups, and thereby derive explicit two-sided linear bounds when the equivariance constraint is lifted.

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Extremal Eigenvalue Problems and Free Boundary Minimal Surfaces in the Ball

Cited by 4 publications

References 40 publications

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

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

Rigidity and vanishing theorems for submanifolds with free boundary in the unit ball

Spectral estimates for free boundary minimal surfaces via Montiel-Ros partitioning methods

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