• Corpus ID: 203902575

Existence and classification of $S^1$-invariant free boundary annuli and Möbius bands in $\mathbb{B}^n$

@article{Fraser2019ExistenceAC,
  title={Existence and classification of \$S^1\$-invariant free boundary annuli and M{\"o}bius bands in \$\mathbb\{B\}^n\$},
  author={Ailana M. Fraser and Pam Sargent},
  journal={arXiv: Differential Geometry},
  year={2019}
}
We explicitly classify all $S^1$-invariant free boundary minimal annuli and Mobius bands in $\mathbb{B}^n$. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for $S^1$-invariant metrics on the annulus and Mobius band. First, we determine the supremum of the $k$-th normalized Steklov eigenvalue among all $S^1$-invariant metrics on the Mobius band for each $k \geq 1$, and show that it is achieved by the induced metric from a free boundary minimal… 

References

SHOWING 1-10 OF 17 REFERENCES
Sharp eigenvalue bounds and minimal surfaces in the ball
We prove existence and regularity of metrics on a surface with boundary which maximize $$\sigma _1 L$$σ1L where $$\sigma _1$$σ1 is the first nonzero Steklov eigenvalue and $$L$$L the boundary length.
Free boundary minimal surfaces of unbounded genus
For each integer $g\geq 1$ we use variational methods to construct in the unit $3$-ball $B$ a free boundary minimal surface $\Sigma_g$ of symmetry group $\mathbb{D}_{g+1}$. For $g$ large, $\Sigma_g$
Extremal problems for Steklov eigenvalues on annuli
We obtain supremum of the $$k$$k-th normalized Steklov eigenvalues of all rotationally symmetric conformal metrics on $$[0,T]\times \mathbb {S}^1$$[0,T]×S1, $$k>1$$k>1. This generalizes the
Some results on higher eigenvalue optimization
  • A. Fraser, R. Schoen
  • Mathematics
    Calculus of Variations and Partial Differential Equations
  • 2020
In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length)
Index of minimal spheres and isoperimetric eigenvalue inequalities
In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres $${\mathbb {S}}^n$$ S n . First, we propose a new approach to
An isoperimetric inequality for Laplace eigenvalues on the sphere
We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the
Some new examples of nonorientable minimal surfaces
The classical Henneberg's minimal surface (1875, [3, 4, 11]) was the unique nonorientable example known until 1981, when Meeks [6] exhibited the first example of a nonorientable, regular, complete,
Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc
Abstract We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid
Free-boundary minimal surfaces with connected boundary in the $3$-ball by tripling the equatorial disc
In the Euclidean unit three-ball, we construct compact, embedded, two-sided free boundary minimal surfaces with connected boundary and prescribed high genus, by a gluing construction tripling the
...
...