Variational aspects of Laplace eigenvalues on Riemannian surfaces

By G. Kokarev. 2011

97 citations recorded by the source corpus.

We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the general regularity properties of $\lambda_k$-extremal metrics and the existence of a partially regular $\lambda_1$-maximiser.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f50c146138397c46c8fa3b0d22aeab303461ceb3

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In Applied Mathematics

1. M. Li
2. R. Petrides
3. R. Schoen
4. A. Fraser
5. Hung Tran
6. Daniel Ketover
7. S. Brendle
8. G. Kokarev

In Geometry and Topology

1. M. Li
2. R. Petrides
3. R. Schoen
4. A. Fraser
5. Hung Tran
6. Daniel Ketover
7. S. Brendle
8. Peter J. McGrath

Citations in the corpus

• Sharp eigenvalue bounds and minimal surfaces in the ball (2012, methodology): “only hope to characterize surfaces up to σ-homothety by conditions on the Steklov spectrum. We will need the following coarse upper bound which combines Theorem 2.3 of [FS] with a bound of G. Kokarev [K]. Theorem 2.2. Let Mbe a compact oriented surface of genus γwith kboundary components. Let σ1 be ”
• Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces (2013, background): “MAXIMAL METRICS FOR THE FIRST EIGENVALUE 33 [27] G. Kokarev.”
• An isoperimetric inequality for Laplace eigenvalues on the sphere (2017, unknown): “A number of important results on the existence and regularity of maximizing metrics has been recently obtained, see [28, 33, 39, 40, 48].”
• Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex boundary (2012, result): “[12], Fraser and Schoen proved that if Σ is a compact orientable surface of genus g with γ boundary components of total length LpBΣq, then σ1pΣqLpBΣq ď 2πpg`γq. Combining this with a bound of Kokarev [20] and Theorem 3.1, we get the following estimate on the boundary length of a minimal surface with”
• SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “10) holds with C = 8π (see [71]).”