Chris Judge

Affine mappings of translation surfaces: geometry and arithmetic

1. Introduction. Translation surfaces naturally arise in the study of billiards in rational polygons (see [ZKa]). To any such polygon P , there corresponds a unique translation surface, S = S(P), such that the billiard flow in P is equivalent to the geodesic flow on S (see, e.g., [Gu2], [Gu3]). There is also a classical relation between translation surfaces and quadratic differentials on a Riemann surface S. Namely, each quadratic differential induces a translation structure on a finite puncturing of S or on a canonical double covering of S. Quadratic differentials have a natural interpretation as cotangent vectors to Teich-müller space, and this connection has proven useful in the study of billiards (see, e.g., [Ma2], [V1]). With a translation surface, S, one associates various algebraic and geometric objects: the induced affine structure of S and the group of affine diffeomorphisms, Aff(S); the holonomy homomorphism, hol : π 1 (S) → R 2 and the holonomy group Hol(S) = hol(π 1 (S)); the flat structure on S and the natural cell decompositions of its metric completion S. In the present paper, we study the relations between these objects, as well as relations among different translation surfaces. Our main focus is the group Aff(S) and the associated group of differentials, (S) ⊂ SL(2, R). The study of these groups began as part of W. Thurstons classification of surface diffeomorphisms in [Th2]. This study continued with the work of W. Veech in [V1] and [V2]. Veech produced explicit examples of translation surfaces S for which (S) is a nonarithmetic lattice. He showed that if (S) is a lattice, then the geodesic flow on S exhibits remarkable dynamical properties. For these reasons, we call (S) the Veech group of S, and if this group is a lattice, then we call S a Veech surface. We now describe the structure of the paper and our main results. In §2, we establish the setting. In particular, we recall the notion of a G-manifold and associated objects: the developing map, the holonomy homomorphism, and the holonomy group. We introduce the notion of the differential of a G-map with respect to a normal subgroup H ⊂ G. We also introduce the spinal triangulation, one of several cell decompositions canonically associated to a flat surface with cone points. In §3, we study G-coverings of G-manifolds. Given such a covering, p : X → Y , we characterize the group …

Determinants of Laplacians and isopolar metrics on surfaces of infinite area

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.

Spectral Simplicity and Asymptotic Separation of Variables

We describe a method for comparing the spectra of two real-analytic families, (at) and (qt), of quadratic forms that both degenerate as a positive parameter t tends to zero. We suppose that the family (at) is amenable to ‘separation of variables’ and that each eigenspace of at is 1-dimensional for some t. We show that if (qt) is asymptotic to (at) at first order as t → 0, then the eigenspaces of (qt) are also 1-dimensional for all but countably many t. As an application, we prove that for the generic triangle (simplex) in Euclidean space (constant curvature space form) each eigenspace of the Laplacian acting on Dirichlet functions is 1-dimensional.

Generic spectral simplicity of polygons

. We study the Laplace operator with Dirichlet or Neumann boundary conditions on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has a simple spectrum. We also address the more general case of geodesic polygons in a constant curvature space form.

The eigenvalues of the Laplacian on domains with small slits

We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of the Laplacian. In particular, we show that as the length of the slit tends to zero, each real-analytic eigenvalue branch tends to an eigenvalue of the original domain. By combining this with our earlier work (arXiv:math/0703616), we obtain the following application: The generic multiply connected polygon has simple spectrum.

Geodesics and nodal sets of Laplace eigenfunctions on hyperbolic manifolds

Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be bounded just in terms of the area of the surface. We also provide constructions of geodesics in hyperbolic surfaces that lie in a nodal set but that do not lie in the fixed point set of a reflection symmetry.