Large values of eigenfunctions on arithmetic hyperbolic surfaces

2013

In Chapter V, with the help of continuity properties of λ i and a construction of P. Buser, we recall a proof of the fact that each λ i is bounded. Then we focus on λ 1 and ask the question if the maximum of λ 1 over M g is more than 1 4 or not. Results due to Burger-Buser-Dodziuk [BBD] and Brooks-Makover [B-M] say that there exist surfaces with λ 1 arbitrary close to 1 4. However, the surfaces constructed in these results are not of the same genus. Using topological arguments, as in Chapter 2, we prove that the answer is yes in the case of genus two i.e. there are surfaces in M 2 such that λ 1 > 1 4. Moreover, we prove that the subset of M 2 containing surfaces with λ 1 > 1 4 disconnects M 2 .

Mathematische Zeitschrift, 2015

We apply topological methods to study the smallest nonzero number λ1 in the spectrum of the Laplacian on finite area hyperbolic surfaces. For closed hyperbolic surfaces of genus two we show that the set {S ∈ M2 : λ1(S) > 1 4 } is unbounded and disconnects the moduli space M2.

Inventiones Mathematicae, 1992

We cite two general questions on the spectrum of the Laplace-Betrami operator for a finite area hyperbolic Riemann surface R. The first is to find the multiplicity of an eigenvalue in the generic case, for both compact and non compact surfaces. The second is to resolve a pair of antipodal conjectures. The Selberg conjecture, that in the Weyl formula for the asymptotics of the spectrum of a surface with cusps, the contribution of the continuous spectrum, the local winding number of the determinant of the scattering matrix, is of lower order of magnitude; thus the discrete spectrum alone would account for the Weyl asymptotics [Hj 1, LP, Sb]. This is the case for congruence subgroups of PSL(2; 2g), [Sb]. On the other hand the Phillips-Sarnak conjecture provides that the generic surface with cusps will have only a finite number of eigenvalues, [PS 1, DIPS]. This is in fact the case for the generic metric with variable negative curvature and standard cusps, [CVI, CV2]. The second conjecture is also supported by the recent numerical investigations of Hejhal, [Hj3].

Inventiones Mathematicae, 1992

We are interested in the limiting behavior of the spectrum of the Laplace-Beltrami operator for a degenerating family of Riemann surfaces with finite area hyperbolic metrics. Our hope is that such an analysis will provide detailed information on the spectrum in general; for instance on the Selberg conjecture, the Phillips-Sarnak conjecture, and on the question of the multiplicity of the general eigenvalue [DIPS, Hjl, PSI, Sb, CV2]. For degenerating finite area surfaces the behavior of eigenvalues in the range [0, 1/4) is relatively well understood [Bul, Bu2, SWY, DPRS, CC, Hjl, Hj3, Ji]. We are now interested in the range (1/4, ~). The eigenvalue 2 = 1/4 is very special. For a noncompact surface the variational behavior of )~ = l/4 is subtle, [PS3]. We consider then a one-parameter degenerating family Rt, 0 < t < 1, and the branch (~t, 2,) of an eigenpair along the family. If the Rt are noncompact then the ~Pt are required to be square-integrable; 2t is in the discrete spectrum of R,. The motivation for considering branches of eigenpairs is simple.

2013

We apply topological methods to study eigenvalues of the Laplacian on closed hyperbolic surfaces. For any closed hyperbolic surface S of genus g, we get a geometric lower bound on λ2g−2(S) : λ2g−2(S)&gt; 1/4+ 0(S), where 0(S)&gt; 0 is an explicit constant which depends only on the systole of S. Introduction. Here a hyperbolic surface is a complete two dimensional Rie-mannian manifold with sectional curvature equal to −1. Any hyperbolic surface is isometric to the quotient H/Γ, where H is the Poincare ́ upper halfplane and Γ is a Fuchsian group, i.e. a discrete torsion-free subgroup of PSL(2,R). The Laplacian on H is the differential operator which associates to a C2- function f the function

Mathematische Annalen, 1993

Journal of Differential Geometry

International Mathematics Research Notices, 2015

Let S be a noncompact, finite area hyperbolic surface of type (g, n). Let ∆S denote the Laplace operator on S. As S varies over the moduli space Mg,n of finite area hyperbolic surfaces of type (g, n), we study, adapting methods of Lizhen Ji [Ji] and Scott Wolpert [Wo], the behavior of small cuspidal eigenpairs of ∆S. In Theorem 2 we describe limiting behavior of these eigenpairs on surfaces Sm ∈ Mg,n when (Sm) converges to a point in Mg,n. Then we consider the i-th cuspidal eigenvalue, λ c i (S), of S ∈ Mg,n. Since non-cuspidal eigenfunctions (residual eigenfunctions or generalized eigenfunctions) may converge to cuspidal eigenfunctions, it is not known if λ c i (S) is a continuous function. However, applying Theorem 2 we prove that, for all k ≥ 2g − 2, the sets C 1 4 g,n(k) = {S ∈ Mg,n : λ c k (S) > 1 4 } are open and contain a neighborhood of ∪ n i=1 M0,3 ∪ Mg−1,2 in Mg,n. Moreover, using topological properties of nodal sets of small eigenfunctions from [O], we show that C 1 4 g,n(2g − 1) contains a neighborhood of M0,n+1 ∪ Mg,1 in Mg,n. These results provide evidence in support of a conjecture of Otal-Rosas [O-R].

Transactions of the American Mathematical Society, 1995

We consider a sequence (M")^=¡ of compact hyperbolic manifolds converging to a complete hyperbolic manifold Mq with cusps. The Laplace operator acting on the space of L2 differential forms on M0 has continuous spectrum filling the half-line [0, oo). One expects therefore that the spectra of this operator on M" accumulate to produce the continuous spectrum of the limiting manifold. We prove that this is the case and obtain a sharp estimate of the rate of accumulation.

Proceedings of the American Mathematical Society

We show a class of perturbations X of the Fermat hypersurface such that any holomorphic curve from ℂ into X is degenerate. Applying this result, we give explicit examples of hyperbolic surfaces in ℙ 3 (ℂ) of arbitrary degree d≥22, and of curves of arbitrary degree d≥19 in ℙ 2 (ℂ) with hyperbolic complements.