Using the maximal regularity theory for quasilinear parabolic systems, we prove two stability results of complex hyperbolic space under the curvature-normalized Ricci flow in complex dimensions two and higher. The first result is on a closed manifold. The second result is on a complete noncompact manifold. To prove both results, we fully analyze the structure of the Lichnerowicz Laplacian on co...

In this note we consider the spectrum of the Laplacian acting on the space of (co-closed) differential forms on the quotient of /i-dimensional hyperbolic space by a co-compact Kleinian group. Using a result of P.-Y. Gaillard we relate these to currents on the sphere at infinity of hyperbolic space with distinctive transformation properties under the action of the group. We analyse these current...

For non-compact hyperbolic surfaces, the appropriate generalization of the eigenvalue spectrum is the resonance set, the set of poles of the resolvent of a meromoprhic continuation of the Laplacian. Hyperbolic surfaces serve as a model case for quantum theory when the underlying classical dynamics is chaotic. In this talk Ill explain how the resonances are defined and discuss our current unders...

This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its “ideal boundary” at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an (n+ 1)-dimensional manifold is the ray [n/4,∞), with no embedded eigenvalues; however, in general there may be discrete eigenvalues below the con...

The flow of a Bingham fluid with inertial terms is simplified into a nonlinear hyperbolic scalar conservation law, regularised by the total variation flow operator (or 1-Laplacian). We give an entropy weak formulation, for which we prove the existence and the uniqueness of the solution. The existence result is established using the convergence of a numerical approximation (a splitting scheme wh...

We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero square-integrable harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on Λ1(M)/Ker(d)?

In this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the “stadium”, convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove existence of a minimzer. We also study some qualitative properties o...

This paper defines a theory of conformal parametrization of digital surfaces made of surfels equipped with a normal vector. The main idea is to locally project each surfel to the tangent plane, therefore deforming its aspect-ratio. It is a generalization of the theory known for polyhedral surfaces. The main difference is that the conformal ratios that appear are no longer real in general. It yi...

For cofinite Kleinian groups (or equivalently, finite-volume three-dimensional hyperbolic orbifolds) with finite-dimensional unitary representations, we evaluate the regularized determinant of the Laplacian using W. Müller’s regularization. We give an explicit formula relating the determinant to the Selberg zeta-function.

We investigate the pointwise and mean square order of the function Z2(s), where Zk(s) = ∫∞ 1 |ζ(12 + ix)|2kx−s dx, k ∈ N. Three conjectures involving Z2(s) and certain exponential sums of Hecke series in short intervals are formulated, which have important consequences in zeta-function theory. A new order result for Z2(s) is obtained, and the function Zk(s) is discussed. 1. Spectral theory and ...