How large can the first eigenvalue be on a surface of genus two
By Dmitry Jakobson, M. Levitin, N. Nadirashvili, Nilima Nigam, Iosif Polterovich. 2005
82 citations recorded by the source corpus.
AbstractSharp upper bounds for the first eigenvalue of the Laplacian on asurface of a fixed area are known only in genera zero and one. Weinvestigate the genus two case and conjecture that the first eigenvalue ismaximized on a singular surface which is realized as a double branchedcovering over a sphere. The six ramification points are chosen in sucha way that this surface has a complex structure of the Bolza surface.We prove that our conjecture follows from a lower bound on the firsteigenvalue of a certain mixed Dirichlet-Neumann boundary value problemon a half-disk. The latter can be studied numerically, and we presentconclusive evidence supporting the conjecture. Keywords: Laplacian, first eigenvalue, surface of genus two, mixed boundaryvalue problem. ∗ Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Str. West,Montreal, QC H3A 2K6, Canada; e-mail jakobson@math.mcgill.ca † Department of Mathematics, Heriot-Watt University, United Kingdom; e-mailM.Levitin@ma.hw.ac.uk
Subfields: Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/54fb83eae1aa3313a440f98bd13d2519b6ab05c8
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In Geometry and Topology
Citations in the corpus
- Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces (2013, background): “Let us mention the sphere (Hersch [22]), the projective plane (Li-Yau [31]), the torus (Girouard [18] and Nadirashvili [36]), the Klein bottle (El Soufi-Giacomini-Jazar [13] and Jakobson-Nadirashvili-Polterovitch [25]), the genus 2 surfaces (Jakobson-Levitin-Nadirashvili-Nigam-Polterovitch [24], Kar”
- Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian (2017, background): “Jakobson-LevitinNadirashvili-Nigam-Polterovich [6] focused their attention to the following metric.”
- SPECTRAL GEOMETRY OF THE STEKLOV PROBLEM (2014, unknown): “Similar existence results have been proved for the first nonzero eigenvalue of the Laplace–Beltrami operator in a fixed conformal class of a closed surface of arbitrary genus, in which case conical singularities have to be allowed (see [65, 89]).”