The first Steklov eigenvalue, conformal geometry, and minimal surfaces

By A. Fraser, R. Schoen. 2009

288 citations recorded by the source corpus.

We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ1L(∂Σ)⩽2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ1(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f14b3a96fcc3332ca93916a7dea8fdc28bf128fb

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Upper bounds for Steklov eigenvalues on surfaces (2012, methodology): “4 combines the methods of [10] and [16].”
SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “[38] Let Ω be a compact Riemannian manifold of dimension n with smooth boundary M .”
Eigenvalue inequalities for mixed Steklov problems (2009, methodology): “…4.1 is quite short: it uses domain monotonicity for eigenvalues of mixed Steklov problems (see sections 3.1 and 3.2), the properties of mixed Steklov eigenvalues for cylindrical domains (see section 2.1), and the classical inequalities between the Dirichlet and Neumann eigenvalues [11, 8, 23].”
Free boundary minimal surfaces in the unit ball : recent advances and open questions (2019, methodology): “As mentioned in the introduction, the subject of free boundary minimal surfaces has attracted a lot of recent attention mostly due to the groundbreaking work of Fraser and Schoen [24] [27] on the extremal Steklov eigenvalue problem on compact surfaces with boundary.”
Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary (2012, methodology): “This result is similar to the classical compactness of minimal surfaces in closed manifolds with positive Ricci curvature by Choi and Schoen [6].”
Index characterization for free boundary minimal surfaces (2016, background): “In fact, it was shown that they are first eigenfunctions [10].”
Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces (2012, unknown): “If such a sequence exists, by inequality (1.3.2) the corresponding genera g n of M n tend to inﬁnity.”
Free boundary minimal surfaces in the unit three-ball via desingularization of the critical catenoid and the equatorial disc (2017, background): “The next non-trivial example is the so-called critical catenoid K (see [9]), which is a catenoid in R3 suitably rescaled so that it meets S2(1) orthogonally.”
Free boundary minimal surfaces in the unit 3-ball (2015, methodology): “Following [2], we refer to such minimal surfaces as free boundary minimal surfaces.”