Sharp asymptotics of the first eigenvalue on some degenerating surfaces

We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial surface has some symmetries. If these symmetries are not present we show that the first eigenvalue normalized by area strictly decreases for the same range of parameters. These results are the main motivation for the construction in [preprint, arXiv:1909.03105v2], where we show a monotonicity result for the normalized first eigenvalue without any symmetry assumptions.

[1]  Albert Y. Zomaya,et al.  Partial Differential Equations , 2007, Explorations in Numerical Analysis.

[2]  Henrik Matthiesen,et al.  Existence of metrics maximizing the first eigenvalue on non-orientable surfaces , 2019, Journal of Spectral Theory.

[3]  Anna Siffert,et al.  Handle attachment and the normalized first eigenvalue. , 2019, 1909.03105.

[4]  S. Nayatani,et al.  Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian , 2017, Comptes Rendus Mathematique.

[5]  R. Petrides.  Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces , 2013, 1310.4697.

[6]  B. Colbois,et al.  Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The `Conformal Spectrum' , 2003, math/0409316.

[7]  O. Post.  Periodic manifolds with spectral gaps , 2002, math-ph/0207017.

[8]  O. Post.  Periodic Manifolds, Spectral Gaps, and Eigenvalues in Gaps , 2000 .

[9]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[10]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[11]  C. Anné.  Fonctions propres sur des variétés avec des anses fines, application à la multiplicité , 1989 .

[12]  C. Anné.  Spectre du laplacien et écrasement d'anses , 1987 .

[13]  Jeffrey Rauch,et al.  Potential and scattering theory on wildly perturbed domains , 1975 .

[14]  Nikolai Nadirashvili,et al.  The Erwin Schrr Odinger International Institute for Mathematical Physics Berger's Isoperimetric Problem and Minimal Immersions of Surfaces Berger's Isoperimetric Problem and Minimal Immersions of Surfaces , 2022 .