arXiv subjects

Colin Guillarmou

Publications and source records attributed to Colin Guillarmou.

At least 55 records · Page 3Linked to original sources

arXiv2016

Marked boundary rigidity for surfaces

We show that, on an oriented compact surface, two sufficiently $C^2$-close Riemannian metrics with strictly convex boundary, no conjugate points, hyperbolic trapped set for their geodesic flows, and same marked boundary distance, are isometric via a diffeomorphism that fixes the boundary. We also prove that the same conclusion holds on a compact surface for any two negatively curved Riemannian metrics with strictly convex boundary and same marked boundary distance, extending a result of Croke and Otal.

math.DG↗

arXiv2016

Pollicott-Ruelle resonances for open systems

We define Pollicott-Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane.

math.DS↗

arXiv2015

Lens rigidity for manifolds with hyperbolic trapped set

For a Riemannian manifold $(M,g)$ with strictly convex boundary $\partial M$, the lens data consists in the set of lengths of geodesics $γ$ with endpoints on $\partial M$, together with their endpoints $(x_-,x_+)\in \partial M\times \partial M$ and tangent exit vectors $(v_-,v_+)\in T_{x_-} M\times T_{x_+} M$. We show deformation lens rigidity for a large class of manifolds which includes all manifolds with negative curvature and strictly convex boundary, possibly with non-trivial topology and trapped geodesics. For the same class of manifolds in dimension $2$, we prove that the set of endpoints and exit vectors of geodesics (ie. the scattering data) determines the topology and the conformal class of the surface.

math.AP↗

arXiv2015

Renormalized volume on the Teichmüller space of punctured surfaces

We define and study the renormalized volume for geometrically finite hyperbolic $3$-manifolds, including with rank-$1$ cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric $g_0$ with rank-$1$ cusps, the renormalized volume converges to the renormalized volume of the limiting metric.

math.DG↗

arXiv2015

Invariant distributions and X-ray transform for Anosov flows

For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $Π$ which maps into the space of invariant distributions in $\cap_{u<0} H^{u}(\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s>0} H^{s}(\mathcal{M})$. We describe relations to Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}=SM$ of a compact manifold, we apply this theory to study questions related to $X$-ray transform on symmetric tensors on $M$: in particular we prove that injectivity implies surjectivity of X-ray transform, and we show injectivity for surfaces.

math.AP↗

arXiv2015

Reconstruction formulas for X-ray transforms in negative curvature

We give reconstruction formulas inverting the geodesic X-ray transform over functions (call it $I_0$) and solenoidal vector fields on surfaces with negative curvature and strictly convex boundary. These formulas generalize the Pestov-Uhlmann formulas in [Pestov-Uhlmann, IMRN '04] (established for simple surfaces) to cases allowing geodesics with infinite length on surfaces with trapping. Such formulas take the form of Fredholm equations, where the analysis of error operators requires deriving new estimates for the normal operator $Π_0 = I_0^* I_0$. Numerical examples are provided at the end.

math.DG↗

arXiv2015

The X-ray transform for connections in negative curvature

We consider integral geometry inverse problems for unitary connections and skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The results apply to manifolds in any dimension, with or without boundary, and also in the presence of trapped geodesics. In the boundary case, we show injectivity of the attenuated ray transform on tensor fields with values in a Hermitian bundle (i.e. vector valued case). We also show that a connection and Higgs field on a Hermitian bundle are determined up to gauge by the knowledge of the parallel transport between boundary points along all possible geodesics. The main tools are an energy identity, the Pestov identity with a unitary connection, which is presented in a general form, and a precise analysis of the singularities of solutions of transport equations when there are trapped geodesics. In the case of closed manifolds, we obtain similar results modulo the obstruction given by twisted conformal Killing tensors, and we also study this obstruction.

math.AP↗

arXiv2015

Power spectrum of the geodesic flow on hyperbolic manifolds

We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.

math.DS↗

arXiv2013

Low energy resolvent for the Hodge Laplacian: Applications to Riesz transform, Sobolev estimates and analytic torsion

On an asymptotically conic manifold $(M,g)$, we analyze the asymptotics of the integral kernel of the resolvent $R_q(k):=(Δ_q+k^2)^{-1}$ of the Hodge Laplacian $Δ_q$ on $q$-forms as the spectral parameter $k$ approaches zero, assuming that 0 is not a resonance. The first application we give is an $L^p$ Sobolev estimate for $d+δ$ and $Δ_q$. Then we obtain a complete characterization of the range of $p>1$ for which the Riesz transform for $q$-forms $T_q=(d+δ)Δ_q^{-1/2}$ is bounded on $L^p$. Finally, we obtain an asymptotic formula for the analytic torsion of a family of smooth compact Riemannian manifolds $(Ω_ε,g_ε)$ degenerating to a compact manifold $(Ω_0,g_0)$ with a conic singularity as $ε\to 0$.

math.AP↗

arXiv2013

Microlocal limits of plane waves and Eisenstein functions

We study microlocal limits of plane waves on noncompact Riemannian manifolds (M,g) which are either Euclidean or asymptotically hyperbolic with curvature -1 near infinity. The plane waves E(z,ξ) are functions on M parametrized by the square root of energy z and the direction of the wave, ξ, interpreted as a point at infinity. If the trapped set K for the geodesic flow has Liouville measure zero, we show that, as z\to +\infty, E(z,ξ) microlocally converges to a measure μ_ξ, in average on energy intervals of fixed size, [z,z+1], and in ξ. We express the rate of convergence to the limit in terms of the classical escape rate of the geodesic flow and its maximal expansion rate - when the flow is Axiom A on the trapped set, this yields a negative power of z. As an application, we obtain Weyl type asymptotic expansions for local traces of spectral projectors with a remainder controlled in terms of the classical escape rate.

math.AP↗

arXiv2013

Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

On geometrically finite hyperbolic manifolds $Γ\backslash H^{d}$, including those with non-maximal rank cusps, we give upper bounds on the number $N(R)$ of resonances of the Laplacian in disks of size $R$ as $R\to \infty$. In particular, if the parabolic subgroups of $Γ$ satisfy a certain Diophantine condition, the bound is $N(R)= O(R^d (\log R)^{d+1})$.

math.SP↗

arXiv2012

The renormalized volume and uniformisation of conformal structures

We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\pl M$ has dimension $n$ even. Its definition depends on the choice of metric $h_0$ on $\partial M$ in the conformal class at infinity determined by $g$, we denote it by ${\rm Vol}_R(M,g;h_0)$. We show that ${\rm Vol}_R(M,g;\cdot)$ is a functional admitting a "Polyakov type" formula in the conformal class $[h_0]$ and we describe the critical points as solutions of some non-linear equation $v_n(h_0)={\rm const}$, satisfied in particular by Einstein metrics. In dimension $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while in dimension $n=4$ this amounts to solving the $σ_2$-Yamabe problem. Next, we consider the variation of ${\rm Vol}_R(M,\cdot;\cdot)$ along a curve of AHE metrics $g^t$ with boundary metric $h_0^t$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_n(h)=\int_{\pl M}v_n(h){\rm dvol}_{h}$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to Identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space $\mc{T}(\pl M)$ of conformal structures on $\pl M$. We obtain as a consequence a higher-dimensional version of McMullen's quasifuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

math.DG↗

arXiv2012

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

For geometrically finite hyperbolic manifolds $Γ\backslash H^{n+1}$, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of $Γ$ in large balls of $H^{n+1}$ in terms of the Hausdorff dimension of the limit set of $Γ$.

math.SP↗

arXiv2012

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.

math.SP↗

arXiv2012

Uniform Sobolev estimates for non-trapping metrics

We prove uniform Sobolev estimates $||u||_{L^{p'}} \leq C ||(Δ-α)u||_{L^{p}}$, where $p=2n/(n+2), p'=2n/(n-2)$, for the Laplacian $Δ$ on non-trapping asymptotically conic manifolds of dimension $n$. Here C is independent of $α$ which ranges over all complex numbers. This generalizes to non-constant coefficient Laplacians a result of Kenig-Ruiz-Sogge.

math.AP↗

arXiv2012

Resolvent at low energy III: the spectral measure

Let $M^\circ$ be a complete noncompact manifold and $g$ an asymptotically conic Riemaniann metric on $M^\circ$, in the sense that $M^\circ$ compactifies to a manifold with boundary $M$ in such a way that $g$ becomes a scattering metric on $M$. Let $Δ$ be the positive Laplacian associated to $g$, and $P = Δ+ V$, where $V$ is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure $dE(λ) = (λ/πi) \big(R(λ+i0) - R(λ- i0) \big)$ of $P_+^{1/2}$, where $R(λ) = (P - λ^2)^{-1}$, as $λ\to 0$, in a manner similar to that done previously by the second author and Vasy, and by the first two authors. The main result is that the spectral measure has a simple, `conormal-Legendrian' singularity structure on a space which is obtained from $M^2 \times [0, λ_0)$ by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators $\cos(t \sqrt{P_+})$ and $\sin(t \sqrt{P_+})/\sqrt{P_+}$, and the Schrödinger propagator $e^{itP}$, as $t \to \infty$. In particular, we prove the analogue of Price's law for odd-dimensional asymptotically conic manifolds. This result on the spectral measure has been used in a follow-up work by the authors (arXiv:1012.3780) to prove sharp restriction and spectral multiplier theorems on asymptotically conic manifolds.

math.AP↗

arXiv2012

Restriction and spectral multiplier theorems on asymptotically conic manifolds

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(λ)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq 2(n+1)/(n+3)$, where $p'$ is the conjugate exponent to $p$, with operator norm scaling as $λ^{n(1/p - 1/p') - 1}$. We prove a geometric generalization in which the Laplacian on $\RR^n$ is replaced by the Laplacian, plus suitable potential, on a nontrapping asymptotically conic manifold, which is the first time such a result has been proven in the variable coefficient setting. It is closely related to, but stronger than, Sogge's discrete $L^2$ restriction theorem, which is an $O(λ^{n(1/p - 1/p') - 1})$ estimate on the $L^p \to L^{p'}$ operator norm of the spectral projection for a spectral window of fixed length. From this, we deduce spectral multiplier estimates for these operators, including Bochner-Riesz summability results, which are sharp for $p$ in the range above.

math.AP↗

arXiv2012

Wave 0-Trace and length spectrum on convex co-compact hyperbolic manifolds

For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $Γ$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. By writing this trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of $Γ$ is greater than $n/2$.

math.DG↗