Sharp eigenvalue bounds and minimal surfaces in the ball
289 citations recorded by the source corpus.
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. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball $$B^n$$Bn for some $$n$$n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in $$B^3$$B3. We also show that the unique solution on the Möbius band is achieved by an explicit $$S^1$$S1 invariant embedding in $$B^4$$B4 as a free boundary surface, the critical Möbius band. For oriented surfaces of genus $$0$$0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in $$B^3$$B3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of $$4\pi $$4π. We also prove multiplicity bounds on $$\sigma _1$$σ1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/b886804f0e48ae3f1ba6b035a14ecdcba6a1565c
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In Applied Mathematics
In Geometry and Topology
Citations in the corpus
- Compactness analysis for free boundary minimal hypersurfaces (2017, background): “In the Euclidean ball, Fraser and Schoen have obtained free boundary minimal surfaces with genus zero and any number of boundary components larger than one [17], while Folha, Pacard and Zolotareva [13] have obtained free boundary minimal surfaces with genus zero or one and any sufficiently large num”
- Index characterization for free boundary minimal surfaces (2016, methodology): “Several existence results were obtained by different methods including various minimax constructions, varifold theory, minimizing the Dirichlet functional and maximizing the first Steklov eigenvalue; see [5, 24, 30, 31, 34, 15, 19, 8, 21, 23, 12, 13] and references therein.”
- Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces (2013, methodology): “This result is similar to the beautiful result concerning Steklov eigenvalues recently obtained by Fraser and Schoen [16].”
- Free boundary minimal surfaces of unbounded genus (2016, methodology): “hout this paper, D will denote the unique free boundary minimal disk in the xy plane, and C the unique critical catenoid that is rotationally symmetric about the z axis. For each k ≥ 2, Fraser-Schoen [FS] constructed a free boundary minimal surface Fk resembling a “doubling” of the flat disk D in the”
- Free boundary minimal surfaces in the unit 3-ball (2015, background): “S2. The fact that the area is critical for variations of the boundary of the surface which are tangent to S2 translates into the fact that the minimal surface meets S2 orthogonally. In a recent paper [3], A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces Σn in B3 wh”
- SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “For instance, Fraser and Schoen [39] proved that on annuli (γ = 0, l = 2), the maximal value of σ1(Ω)L(∂Ω) is attained by the critical catenoid (σ1L ∼ 4π/1.”
- Variational aspects of Laplace eigenvalues on Riemannian surfaces (2011, methodology): “First, the results in our Example 1.3 (the Steklov eigenvalueproblem) have been independently obtained in [7]. Extremal problems for Steklov eigenvalues have been also studied by Fraser and Schoen in [14] where the authors prove the existence of a λ1-maximiser for zero genus surfaces. The state of t”
- A Characterization of the Critical Catenoid (2016, background): “[6] A. Fraser and R. Schoen, Sharp eigenvalue bounds and minimal surfaces in the ball, Invent. math.”
- Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary (2012, background): “operly embedded minimal surfaces in the unit ball B3 with free boundary? Since B3 is simply connected, any properly embedded surface in B3 must be orientable. By the results of Fraser and Schoen [10] [11], we know any genus zero orientable surface can be minimally embedded into B3 as a free boundary”