Compact Hyperbolic Surface Research Articles

Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

We consider the action of SL(2, R) on a vector bundle H preserving an ergodic probability measure nu on the base X. Under an irreducibility assumption on this action, we prove that if (nu) over cap is any lift of nu to a probability measure on the projectivized bunde P(H) that is invariant under the upper triangular subgroup, then (nu) over cap is supported in the projectivization P(E-1) of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [57]. Second, if P(V) is an irreducible, flat projective bundle over a compact hyperbolic surface Sigma, with hyperbolic foliation F tangent to the flat connection, then the foliated horocycle flow on (TF)-F-1 is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [13] to arbitrary dimension.

A central limit theorem for random closed geodesics: Proof of the Chas–Li–Maskit conjecture

We prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface. In the special case of a hyperbolic pair of pants, this settles a conjecture of Chas–Li–Maskit.

Bergman kernel on Riemann surfaces and Kähler metric on symmetric products

Let [Formula: see text] be a compact hyperbolic Riemann surface equipped with the Poincaré metric. For any integer [Formula: see text], we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle [Formula: see text], where [Formula: see text] is the holomorphic cotangent bundle of [Formula: see text]. Our first main result estimates the corresponding Bergman metric on [Formula: see text] in terms of the Poincaré metric. We then consider a certain natural embedding of the symmetric product of [Formula: see text] into a Grassmannian parametrizing subspaces of fixed dimension of the space of all global holomorphic sections of [Formula: see text]. The Fubini–Study metric on the Grassmannian restricts to a Kähler metric on the symmetric product of [Formula: see text]. The volume form for this restricted metric on the symmetric product is estimated in terms of the Bergman kernel of [Formula: see text] and the volume form for the orbifold Kähler form on the symmetric product given by the Poincaré metric on [Formula: see text].

Bounds for fixed points on products of hyperbolic surfaces

For the product $S_1\times S_2$ of any two connected compact hyperbolic surfaces $S_1$ and $S_2$, we give a finite bound $\mathcal{B}$ such that for any self-homeomorphism $f$ of $S_1\times S_2$ and any fixed point class $F$ of $f$, the index $|ind(f, F)|\leq \mathcal{B}$, which is an affirmative answer for a special case of a question asked by Boju Jiang. Moreover, we also give bounds for the Lefschetz number $L(f)$ and the Nielsen number $N(f)$ of the homeomorphism $f$.

Unique ergodicity of the horocycle flow on Riemannnian foliations

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.

Inner Product of Eigenfunctions over Curves and Generalized Periods for Compact Riemannian Surfaces

We show that for a smooth closed curve $$\gamma $$ on a compact Riemannian surface without boundary, the inner product of two eigenfunctions $$e_\lambda $$ and $$e_\mu $$ restricted to $$\gamma $$ , $$|\int e_\lambda \overline{e_\mu }\,\text {d}s|$$ , is bounded by $$\min \{\lambda ^\frac{1}{2},\mu ^\frac{1}{2}\}$$ . Furthermore, given $$0<c<1$$ , if $$0<\mu <c\lambda $$ , we prove that $$\int e_\lambda \overline{e_\mu }\,\text {d}s=O(\mu ^\frac{1}{4})$$ , which is sharp on the sphere $$S^2$$ . These bounds unify the period integral estimates and the $$L^2$$ -restriction estimates in an explicit way. Using a similar argument, we also show that the $$\nu $$ th order Fourier coefficient of $$e_\lambda $$ over $$\gamma $$ is uniformly bounded if $$0<\nu <c\lambda $$ , which generalizes a result of Reznikov for compact hyperbolic surfaces, and is sharp on both $$S^2$$ and the flat torus $$\mathbb T^2$$ . Moreover, we show that the analogs of our results also hold in higher dimensions for the inner product of eigenfunctions over hypersurfaces.

Supnorm of an eigenfunction of finitely many Hecke operators

Let $\phi$ be a Laplace eigenfunction on a compact hyperbolic surface attached to an order in a quaternion algebra. Assuming that $\phi$ is an eigenfunction of Hecke operators at a \emph{fixed finite} collection of primes, we prove an $L^\infty$-norm bound for $\phi$ that improves upon the trivial estimate by a power of the logarithm of the eigenvalue. We have constructed an amplifier whose length depends on the support of the amplifier on Hecke trees. We have used a method of B\'erard in \cite{Be} to improve the archimedean amplification.

Kähler Geometry on Hurwitz Spaces

The classical Hurwitz space \mathcal{H}^{n,b} is a fine moduli space for simple branched coverings of the Riemann sphere \mathbb{P}^1 by compact hyperbolic Riemann surfaces. In the article we study a generalized Weil-Petersson metric on the Hurwitz space, which was introduced in [R. Axelsson et al., Manuscr. Math. 147, No. 1–2, 63–79 (2015; Zbl 1319.32012)]. For this purpose, Horikawa's deformation theory of holomorphic maps is refined in the presence of hermitian metrics in order to single out distinguished representatives. Our main result is a curvature formula for a subbundle of the tangent bundle on the Hurwitz space obtained as a direct image. This covers the case of the curvature of the fibers of the natural map \mathcal{H}^{n,b} \to \mathcal{M}_g .

Joining measures for horocycle flows on abelian covers

A celebrated result of Ratner from the eighties says that two horocycle flows on hyperbolic surfaces of finite area are either the same up to algebraic change of coordinates, or they have no non-trivial joinings. Recently, Mohammadi and Oh extended Ratner's theorem to horocycle flows on hyperbolic surfaces of infinite area but finite genus. In this paper, we present the first joining classification result of a horocycle flow on a hyperbolic surface of infinite genus: a $\mathbb{Z}$ or $\mathbb{Z}^2$-cover of a general compact hyperbolic surface.

Extremal disc packings in compact hyperbolic surfaces

We study packings of metric discs with respect to the canonical hyperbolic metric of a compact Riemann surface of genus greater than one. We find the maximum radius of a packing as a function of the genus and the number of discs and we investigate some properties of the surfaces that contain an extremal packing.

Quantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaces

We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdi\`{e}re. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and the first-named author. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Lindenstrauss and the first-named author on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalised averaging operators over discs, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.

Fuchsian groups and compact hyperbolic surfaces

We present a topological proof of the following theorem of Benoist-Quint: for a finitely generated non-elementary discrete subgroup \Gamma_1 of PSL (2, \mathbb R) with no parabolics, and for a cocompact lattice \Gamma_2 of PSL (2, \mathbb R) , any \Gamma_1 orbit on \Gamma_2 \ PSL (2, \mathbb R) is either finite or dense.

The equivariant Minkowski problem in Minkowski space

The classical Minkowski problem in Minkowski space asks, for a positive function $\phi$ on $\mathbb{H}^d$, for a convex set $K$ in Minkowski space with $C^2$ space-like boundary $S$, such that $\phi(\eta)^{-1}$ is the Gauss--Kronecker curvature at the point with normal $\eta$. Analogously to the Euclidean case, it is possible to formulate a weak version of this problem: given a Radon measure $\mu$ on $\mathbb{H}^d$ the generalized Minkowski problem in Minkowski space asks for a convex subset $K$ such that the area measure of $K$ is $\mu$. In the present paper we look at an equivariant version of the problem: given a uniform lattice $\Gamma$ of isometries of $\mathbb{H}^d$, given a $\Gamma$ invariant Radon measure $\mu$, given a isometry group $\Gamma_{\tau}$ of Minkowski space, with $\Gamma$ as linear part, there exists a unique convex set with area measure $\mu$, invariant under the action of $\Gamma_{\tau}$. The proof uses a functional which is the covolume associated to every invariant convex set. This result translates as a solution of the Minkowski problem in flat space times with compact hyperbolic Cauchy surface. The uniqueness part, as well as regularity results, follow from properties of the Monge--Amp\`ere equation. The existence part can be translated as an existence result for Monge--Amp\`ere equation. The regular version was proved by T.~Barbot, F.~B\'eguin and A.~Zeghib for $d=2$ and by V.~Oliker and U.~Simon for $\Gamma_{\tau}=\Gamma$. Our method is totally different. Moreover, we show that those cases are very specific: in general, there is no smooth $\Gamma_\tau$-invariant surface of constant Gauss-Kronecker curvature equal to $1$.

Improved Critical Eigenfunction Restriction Estimates on Riemannian Surfaces with Nonpositive Curvature

We show that one can obtain improved L 4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge’s strategy in (Improved critical eigenfunction estimates on manifolds of nonpositive curvature, Preprint). We first combine the improved L 2 restriction estimate of Blair and Sogge (Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint) and the classical improved $${L^\infty}$$ estimate of Bérard to obtain an improved weak-type L 4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger (Math Res Lett 18(4):767–781, 2011). This estimate improves the L 4 restriction estimate of Burq et al. (Duke Math J 138:445–486, 2007) and Hu (Forum Math 6:1021–1052, 2009) by a power of $${(\log\log\lambda)^{-1}}$$ . Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of $${(\log\lambda)^{-1}}$$ by applying the ideas from (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) and (Blair and Sogge, Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint). We are able to compute various constants that appeared in (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover $${\mathbb{H}^2}$$ .

Refinement of prime geodesic theorem

We prove existence of a set $E$ of positive real numbers, which is relatively small in the sense that its logarithmic measure is finite, such that we can improve the error term of the prime geodesic theorem as $x\to\infty$ $(x\notin E)$. The result holds for any compact hyperbolic surfaces, and it would also be true for generic hyperbolic surfaces of finite volume according to the philosophy of Phillips and Sarnak.

Compact non-orientable surfaces of genus 6 with extremal metric discs

A compact hyperbolic surface of genus g g is said to be extremal if it admits an extremal disc, a disc of the largest radius determined only by g g . We discuss how many extremal discs are embedded in non-orientable extremal surfaces of genus 6. This is the final genus in our interest because it is already known for g = 3 , 4 , 5 g=3, 4, 5 , or g &gt; 6 g&gt;6 . We show that non-orientable extremal surfaces of genus 6 admit at most two extremal discs. The locus of extremal discs is also obtained for each surface. Consequently non-orientable extremal surfaces of arbitrary genus g ≧ 3 g\geqq 3 admit at most two extremal discs. Furthermore we determine the groups of automorphisms of non-orientable extremal surfaces of genus 6 with two extremal discs.

Construction of hyperbolic Riemann surfaces with large systoles

Let S be a compact hyperbolic Riemann surface of genus \({g \geq 2}\). We call a systole a shortest simple closed geodesic in S and denote by \({{\rm sys}(S)}\) its length. Let \({{\rm msys}(g)}\) be the maximal value that \({{\rm sys}(\cdot)}\) can attain among the compact Riemann surfaces of genus g. We call a (globally) maximal surface Smax a compact Riemann surface of genus g whose systole has length \({{\rm msys}(g)}\). In Section 2 we use cutting and pasting techniques to construct compact hyperbolic Riemann surfaces with large systoles from maximal surfaces. This enables us to prove several inequalities relating \({{\rm msys}(\cdot)}\) of different genera. In Section 3 we derive similar intersystolic inequalities for non-compact hyperbolic Riemann surfaces with cusps.

Logarithmic-Scale Quasimodes that do not Equidistribute

Given any compact hyperbolic surface $M$, and a closed geodesic on $M$, we construct of a sequence of quasimodes on $M$ whose microlocal lifts concentrate positive mass on the geodesic. Thus, the Quantum Unique Ergodicity (QUE) property does not hold for these quasimodes. This is analogous to a construction of Faure-Nonnenmacher-De Bi\`evre in the context of quantized cat maps, and lends credence to the suggestion that large multiplicities play a role in the known failure of QUE for certain toy models of quantum chaos. We moreover conjecture a precise threshold for the order of quasimodes needed for QUE to hold--- the result of the present paper shows that this conjecture, if true, is sharp.

Bounds for Green’s functions on noncompact hyperbolic Riemann orbisurfaces of finite volume

In Jorgenson and Kramer (Compos Math 142:679–700, 2006) derived bounds for the canonical Green’s function and the hyperbolic Green’s function defined on a compact hyperbolic Riemann surface. In this article, we extend these bounds to noncompact hyperbolic Riemann orbisurfaces of finite volume and of genus greater than zero, which can be realized as a quotient space of the action of a Fuchsian subgroup of first kind on the hyperbolic upper half-plane.

Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces

This article is a first step towards the understanding of the dynamics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Martínez and A. Verjovsky on the minimality of this flow assuming that the “natural” affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G.A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of PSL(2,R) and a Lie group G, which places us within the homogeneous framework investigated by M. Ratner. Since our purpose is to deal with non-homogeneous situations, we do not use Ratner’s famous Orbit-Closure Theorem, but we give an elementary proof. We show that this special situation arises for homogeneous Riemannian and Lie foliations, reintroducing the foliation point of view. Examples and counter-examples take an important place in our work, in particular, the very instructive case of the solvable manifold TA3. Our aim in writing this text is to offer to the reader an accessible introduction to a subject that was intensively studied in the algebraic setting, although there still are unsolved geometric problems.