Mini-conference ``Géométrie et analyse sur les variétés riemanniennes non-compactes''

Mini-conference ``Geometry and analysis on non-compact Riemannian manifolds''

La conférence commencera le mercredi 18 septembre à 14h, elle se terminera le samedi 21 à 12h30. Les exposés auront tous lieu à l'Université Montpellier II, Place Eugene Bataillon, batiment de maths (numéro 9).

L'accueil des participants se fera le mercredi 18 septembre à 14h dans le jardin le long du batiment de maths, ou si le temps ne le permet pas, dans la salle de réunion au deuxième étage.

The conference will begin on Wednesday, September 18th, at 2 pm, and it will end on Saturday, september 21st, around 12:30. All talks will take place at the math building of University Montpellier II, Place Eugene Bataillon in Montpellier.

Participants will be welcomed on Wednesday, September 18th at 2 pm in the garden close to the math building (building number 9), or, if the weather is too bad, in the meeting room of the math department, at the second floor of building number 9.

Les inscriptions sont toujours ouvertes, envoyez un message à l'un des organisateurs. Après le 31 Aout 2002, les nouveaux participants seront toujours les bienvenus mais nous ne pourrons plus leur reserver un hébergement à Montpellier ; les participants non-inscrits à cette date devront donc réserver par eux-memes un hotel sur Montpellier.

Inscriptions to the conference are still open ; please send an e-mail to one of the organisers. Note that new participants will always be welcomed, but that we will not be able to make any new hotel reservations after August 31st. New particpants that have not contacted the organisers on september 1st must find a hotel by themselves.

Il n'y a pas de frais d'inscription à payer.

There are no conference fees.

 

A Bientot ! See you soon

Toutes les conférences auront lieu au bâtiment de Mathématiques de l'Université Montpellier II (bât. 9). All talks will take place at the Math Building (building $ \sharp\, 9$) at Montpellier II University.

Pour aller à l'Université, prendre le tramway, direction Mosson, et descendre à la station Université des Sciences et des Lettres. Prendre alors à droite (en tête de quai) l'avenue A. Fliche, le long de l'hôpital Guy-de-Chauliac, en direction d'une sculpture rouge très visible représentant un beignet surmonté d'une saucisse (sic !). Vous trouverez l'accès principal de l'Université immédiatement après la sculpture. Entrez et prenez la route de droite, le bâtiment de mathématiques (numéro 9) se situe à environ 250 mètres de l'entrée principale sur cette route (voir le plan du campus).

To go the University, take the tramway line towards Mosson, get off at Université des Sciences et des Lettres. Take then the street on your right (avenue A. Fliche), and walk along Guy-de-Chauliac hospital until you reach a clearly visible red sculpture picturing a sausage on a doughnut (yes !). The University's main entrance is located a few meters after the scuplture. Get inside and take the main road on the right. The Math building (building number 9) is to be found on this road about 250 metres after the entrance (see the campus map).

Programme

 

Mercredi 18 Septembre 2002

Salle de réunion (deuxième étage, bât. 9) / meeting room (second floor, math building)

14 h -- Accueil des participants.

Salle 331 (troisième étage, bât. 9) / lecture room 331 (third floor, building $ \sharp\, 9$)

15 h -- Mini-cours/Lecture series : Laurent SALOFF-COSTE (Cornell University)

Sobolev inequalities and geometry, I.

16 h -- Pause café / Coffee break

16 h 15 -- Frank PACARD (Paris XII)

Connected sums for Poincaré-Einstein metrics.

17 h 15 --- Thérèse FALLIERO (Avignon)

Hyperbolic and horocyclic Eisenstein series on a non compact Riemann surface.

 

Jeudi 19 Septembre 2002

Salle 331 (troisième étage, bât. 9) / lecture room 331 (third floor, building $ \sharp\, 9$)

9 h 30 -- Mini-cours/Lecture series : Laurent SALOFF-COSTE (Cornell University)

Sobolev inequalities and geometry, II.

10 h 30 -- Pause café / Coffee break

11h -- Hélène DAVAUX (Montpellier)
About scalar curvature and spectrum.

12h -- Déjeuner / Lunch.

14 h 30 -- Nader YEGANEFAR (Nantes)

$ L^2$-cohomology of negatively curved manifolds of finite volume.

15 h 20 -- Pause café / Coffee break

15 h 40 -- Philippe CASTILLON (Montpellier)
A spectral inverse problem on surfaces.

16 h 40 -- Erwann DELAY (Tours)
Prescribing scalar curvature after Corvino and Schoen.

20h -- Dîner (sous réserves) / Conference dinner (to be confirmed).

Vendredi 20 Septembre 2002

Salle 331 (troisième étage, bât. 9) / lecture room 331 (third floor, building $ \sharp\, 9$)

9 h 30 -- Mini-cours/Lecture series : Laurent SALOFF-COSTE (Cornell University)
Sobolev inequalities and geometry, III.

10 h 30 -- Pause café / Coffee break

11 h -- Hervé PAJOT (Cergy-Pontoise)
$ l_{p}$-cohomology, Poincaré inequalities and Besov spaces.

12h -- Déjeuner / Lunch. 0

14 h 30 -- Marc BOURDON (Lille)
$ l_{p}$-cohomology and quasi-isometries of some amalgamated products.

15 h 20 -- Pause café / Coffee break

15 h 40 -- Thierry COULHON (Cergy-Pontoise)
Riesz transforms, Littlewood-Paley-Stein functions and heat kernels on Riemannian non-compact manifolds.

16 h 40 -- A préciser / To be announced

Samedi 21 Septembre 2002

Salle 32 (rez-de-chaussée, bât. 9) / lecture room 32 (ground floor, building $ \sharp\, 9$)

9 h 30 -- Erwann AUBRY (Grenoble)
Estimates on eigensections for Schrödinger operators.

10 h 20 -- Pause café / Coffee break

10 h 35 -- Gilles COURTOIS (Palaiseau)

Titre à préciser /To be announced.

11 h 30 -- Jochen BRÜNING (Berlin)
Dirac systems.

12 h 15 -- Fin de la mini-conférence/End of the mini-conference.

Mini-cours / Lecture series

Laurent SALOFF-COSTE (Cornell University)

Sobolev inequalities and geometry.

(3 $ \times$ 1 h.)

* * *

Résumés des conférences / Abstracts of the talks

(50 min.)
Erwann AUBRY (Grenoble) Estimates on eigensections for Schrödinger operators.
.

Marc BOURDON (Lille) $ l_{p}$-cohomology and quasi-isometries of some amalgamated products.
We use the first $ l_{p}$-cohomology group to study quasi-isometries of some amalgamated products and we deduce rigidity for quasi-isometries of some decompositions into amalgamated products.

Jochen BRÜNING (Berlin) Dirac systems.
We propose a simple functional analytical model which describes Dirac type operators near certain isolated singularities, for complete as well as for noncomplete metrics. Then from relatively simple and natural axioms for the model, we can derive substantial information on the spectral properties of the operators in question. These comprise a description of the essential spectrum, index properties, and asymptotic expansions of the (local) resolvant trace.
This is, to a large extent, a report on jointwork with Werner Ballmann and Gilles Carron.

Philippe CASTILLON (Montpellier) A spectral inverse problem on surfaces.
The purpose of this talk is to see how the geometry of a Riemannian surface can be related to the positivity of some operators on this surface (the operators considered here are of the form $ \Delta+\lambda K$ where $ \Delta$ is the Laplacian of the surface, $ K$ is its curvature and $ \lambda$ is a real number). This problem has its origin in the study of stable minimal surfaces.

Thierry COULHON (Cergy-Pontoise) Riesz transform, Littlewood-Paley-Stein functions and heat kernels on non-compact Riemannian manifolds.
This talk reports on joint work with Xuan Thinh Duong. Robert Strichartz has asked in 1983 for which complete non-compact Riemannian manifolds $ M$ and which $ p\in]1,+\infty[$ one has

$\displaystyle C_p^{-1}\Vert\Delta^{1/2} f\Vert _p\le \Vert\vert\nabla f\vert\V... ..._p\le C_p \Vert\Delta^{1/2} f\Vert _p,\ \forall\,f\in\mathcal{C}_0^\infty(M). $

The second inequality above means that the Riesz transform is bounded on $ L^p(M)$. We proved in 1999 that the Riesz transform is bounded on $ L^p(M)$, $ 1< p\le 2$, and has weak type $ (1,1)$ if $ V(x,2r)\le C\,V(x,r),\ \forall\,x\in M,\,r>0$ and $ p_t(x,x)\le \frac{C}{V(x,\sqrt{t})},\ \forall\,x\in M,\,t>0$, where $ V(x,r)$ is the Riemannian volume of the geodesic ball of center $ x\in M$ and radius $ r>0$, and $ p_t(x,y)$, $ t>0$, $ x,y\in M$ is the heat kernel on $ M$. The example of two Euclidean planes glued by a cylinder shows that additional assumptions are needed in the case $ p>2$. We give several positive and negative results in this direction.

Gilles COURTOIS (Palaiseau) To be announced.

Hélène DAVAUX (Montpellier) About scalar curvature and spectrum.
For a Riemannian closed (compact, without boundary) spin manifold $ (M,g)$ and under some topological assumption (non-zero $ \widehat{A}$-genus or enlargeability in the sense of Gromov-Lawson), we give an optimal upper bound for the infimum of the scalar curvature in terms of the first eigenvalue of the Laplacian on the universal covering $ \widetilde{M}$. More precisely,

$\displaystyle \lambda_0(\widetilde{M}, \widetilde{g}) \leq - \frac{1}{4} \frac{n-1}{n} \inf_M \textrm{scal}(M,g).$

This work improves an inequality which was first proved by K. Ono in $ 1988$.

Erwann DELAY (Tours) Prescribing scalar curvature after Corvino and Schoen.
On a Riemannian manifold $ (M,g_0)$ with scalar curvature $ S_0$, let us give a function $ S$ equal to $ S_0$ outside a compact set $ K$. We construct a metric on $ M$ which coincide with $ g_0$ outside $ K$ and with scalar curvature $ S$. The method used (originally due to Corvino and Schoen) allows to construct by gluing a large class of asymptotically flat manifold with vanishing scalar curvature which are exactly Schwarzschild outside a compact set. This has non trivial implications in General Relativity.

Thérèse FALLIERO (Avignon) Horocyclic and hyperbolic Eisenstein series on a non compact Riemann surface.
We consider M a hyperbolic Riemann surface of finite volume and non compact. First of all, we give the definition and properties of horocyclic Eisenstein series, used in particular in the spectral decomposition of a 1-form. Then we recall the construction of hyperbolic Eisenstein series given by Kudla and Millson in "Harmonic differentials and closed geodesics on a Riemann surface" (Invent. Math., 54, 1979). This enables to obtain the harmonic differential dual to a simple closed geodesic on M. In the non compact case, there is moreover the question of the harmonic differential dual to an infinite geodesic joining two punctures on M. We solve it with two methods: adapting that of Kudla and Millson or using degeneration of Riemann surfaces.

Frank PACARD (Paris XII) Connected sums for Poincaré-Einstein metrics.
I will explain how to ``connect at infinity" two conformally compact manifolds which are Einstein and produce another conformally compact Einstein manifold.

Hervé PAJOT (Cergy-Pontoise) $ l_{p}$-cohomology, Poincaré inequalities and Besov spaces.
Let $ Z$ be a compact metric space. Under some natural assumptions, we define the $ l_{p}$-cohomology (denoted by $ l_{p} H^{.} (J)$) of $ Z$, or more precisely of a family of metrics $ J$ of $ Z$ called the conformal gauge of $ Z$. We show that for any Ahlfors-regular metric $ d$ in $ J$, $ l_{p} H^{1} (J)$ is isomorphic to the $ p$-Besov space of $ (Z,d)$. As an application, when $ J$ contains a Loewner metric (that is a metric $ d$ such that $ (Z,d)$ supports a Poincaré inequality), the critical $ l_{p}$ dimension of $ Z$ is equal to the Pansu conformal dimension. We deduce from this result that in general the conformal gauge of the boundary of a Gromov hyperbolic group does not contain a Loewner metric. This result is related to J. Cannon's conjecture concerning the hyperbolization of $ 3$-manifolds (joint work with Marc Bourdon).

Nader YEGANEFAR (Nantes) $ L^2$-cohomology of negatively curved manifolds of finite volume.
We give a topological interpretation of the space of $ L^2$ harmonic forms on manifolds of finite volume and sufficiently pinched negative curvature. We first introduce notions about $ L^2$-cohomology. Then we show how a hypothesis on the spectrum of a manifold allows us to obtain links between $ L^2$-cohomology, topology and geometry at infinity. We finally apply these results to the particular case of interest.

2002-09-11