Polyakov’s formulation of $2d$ bosonic string theory

Publications mathématiques de l'IHÉS -     

Strichartz Estimates Without Loss on Manifolds with Hyperbolic Trapped Geodesics

In [Do], Doi proved that the ${L^{2}_{t}H^{1/2}_{x}}In [Do], Doi proved that the L²_tH^1/2_x{L^{2}_{t}H^{1/2}_{x}} local smoothing effect for Schr?dinger equations on a Riemannian manifold does not hold if the geodesic flow has one trapped trajectory. We show in contrast that Strichartz estimates and L ¹ → L ^∞ dispersive estimates still hold without loss for e ^itΔ in various situations where the trapped set is hyperbolic and of sufficiently small fractal dimension.     

Random Walk on Surfaces with Hyperbolic Cusps

We consider the operator associated with a random walk on finite volume surfaces with hyperbolic cusps. We study the spectral gap (upper and lower bound) associated with this operator and deduce some rate of convergence of the iterated kernel towards its stationary distribution.     

Bergman and Calderón Projectors for Dirac Operators

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$ , we give a new construction of the Calderón projector on $\partial\bar{X}$ and of the associated Bergman projector on the space of L ² harmonic spinors on $\bar{X}$ , and we analyze their Schwartz kernels. Our approach is based on the conformal covariance of $D_{\bar{g}}$ and the scattering theory for the Dirac operator associated with the complete conformal metric $g=\bar{g}/\rho^{2}$ where ρ is a smooth function on $\bar{X}$ which equals the distance to the boundary near $\partial\bar{X}$ . We show that $\frac{1}{2}(\operatorname{Id}+\tilde{S}(0))$ is the orthogonal Calderón projector, where $\tilde{S}(\lambda)$ is the holomorphic family in {?(λ)≥0} of normalized scattering operators constructed in Guillarmou et al. (Adv. Math., 225(5):2464–2516, 2010), which are classical pseudo-differential of order 2λ. Finally, we construct natural conformally covariant odd powers of the Dirac operator on any compact spin manifold.     

Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I.

Let $M^\circLet be a complete noncompact manifold and g an asymptotically conic manifold on , in the sense that compactifies to a manifold with boundary M in such a way that g becomes a scattering metric on M. A special case that we focus on is that of asymptotically Euclidean manifolds, where the induced metric at infinity is equal to the standard metric on S ⁿ⁻¹; such manifolds have an end that can be identified with in such a way that the metric is asymptotic in a precise sense to the flat Euclidean metric. We analyze the asymptotics of the resolvent kernel (P + k ²)⁻¹ where is the sum of the positive Laplacian associated to g and a real potential function which vanishes to second order at the boundary (i.e. decays to second order at infinity on ) and such that if . Then we show that on a blown up version of the resolvent kernel is conormal to the lifted diagonal and polyhomogeneous at the boundary, and we are able to identify explicitly the leading behaviour of the kernel at each boundary hypersurface. Using this we show that the Riesz transform of P is bounded on for 1 < p < n if , and that this range is optimal if or if M has more than one end. The result with is new even when , g is the Euclidean metric and V is compactly supported. When V ≡ 0 with one end, the range of p becomes 1 <  p <  p _max where p _max > n depends explicitly on the first non-zero eigenvalue of the Laplacian on the boundary . Our results hold for all dimensions ≥ 3 under the assumption that P has neither zero modes nor a zero-resonance. In the follow-up paper Guillarmou and Hassell (Resolvent at low energy and Riesz transform for Schr?dinger operators on asymptotically conic manifolds, preprint) [7] we analyze the same situation in the presence of zero modes and zero-resonances.     

The Linearized Calderón Problem on Complex Manifolds

In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact subdomains of Stein manifolds and sufficiently small subdomains of Kähler manifolds. Some of these manifolds do not admit limiting Carleman weights, and thus cannot be treated by standard methods for the Calderón problem in higher dimensions. The argument is based on constructing Morse holomorphic functions with approximately prescribed critical points. This extends earlier results from the case of Riemann surfaces to higher dimensional complex manifolds.     

Scattering Phase Asymptotics with Fractal Remainders

For a Riemannian manifold (M, g) which is isometric to the Euclidean space outside of a compact set, and whose trapped set has Liouville measure zero, we prove Weyl type asymptotics for the scattering phase with remainder depending on the classical escape rate and the maximal expansion rate. For Axiom A geodesic flows, this gives a polynomial improvement over the known remainders. We also show that the remainder can be bounded above by the number of resonances in some neighbourhoods of the real axis, and provide similar asymptotics for hyperbolic quotients using the Selberg zeta function.     

Resonances on Some Geometrically Finite Hyperbolic Manifolds

Abstract

We first prove the meromorphic extension to ? for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the quotients Γ\ ⁿ⁺¹ with rational nonmaximal rank cusps previously studied by Froese-Hislop-Perry.     

Inverse Scattering at Fixed Energy on Surfaces with Euclidean Ends

On a fixed Riemann surface (M ₀, g ₀) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix S _V (λ) at frequency λ > 0 for the operator Δ+V determines the potential V if \({V\in C^{1,\alpha}(M_0)\cap e^{-\gamma d(\cdot,z_0)^j}L^\infty(M_0)}\) for all γ > 0 and for some \({j\in\{1,2\}}\) , where d(z, z ₀) denotes the distance from z to a fixed point \({z_0\in M_0}\) . The topological condition is given by \({N\geq \max(2g+1,2)}\) for j = 1 and by N ≥ g + 1 if j = 2. In \({\mathbb {R}^2}\) this implies that the operator S _V (λ) determines any C ^{1, α} potential V such that \({V(z)=O(e^{-\gamma|z|^2})}\) for all γ > 0.