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Actes de la Table ronde de geometrie differentielle: En by Arthur L. Besse (Ed.)

By Arthur L. Besse (Ed.)

Résumé :
En juillet 1992, une desk Ronde de Géométrie Différentielle s'est tenue au CIRM de Luminy en l'honneur de Marcel Berger. Les conférences qui sont reproduites dans ces Actes recouvrent los angeles plupart des sujets abordés par Marcel Berger en Géométrie Différentielle et plus précisément : l'holonomie (Bryant), los angeles courbure [courbure sectionnelle optimistic (Grove), courbure sectionnelle négative (Abresch et Schroeder, Ballmann et Ledrappier), courbure de Ricci négative (Lohkamp), courbure scalaire (Delanoë, Hebey et Vaugon), courbure totale (Shioya)], le spectre du laplacien (Anné, Colin de Verdière, Matheus, Pesce), les inégalités isopérimétriques et les systoles (Calabi, Carron, Gromov), ainsi que quelques sujets annexes [espaces d'Alexandrov (Shiohama et Tanaka, Yamaguchi), elastica (Koiso), géométrie sous-riemannienne (Valère et Pelletier)]. Les auteurs sont pour los angeles plupart des géomètres confirmés, dont plusieurs ont travaillé avec Marcel Berger, mais aussi quelques jeunes. Plusieurs articles (Bryant, Colin, Grove...) contiennent une présentation synthétique des résultats récents dans le domaine concerné, pour mieux les rendre obtainable à un public de non-spécialistes.

Abstract:
Proceedings of the around desk in Differential Geometry in honour of Marcel Berger
July 1992, a around desk in Differential Geometry used to be prepared on the CIRM in Luminy (France) in honour of Marcel Berger. In those lawsuits, contributions hide many of the fields studied by means of Marcel Berger in Differential Geometry, particularly : holonomy (Bryant), curvature [positive sectional curvature (Grove), adverse sectional curvature (Abresch and Schroeder, Ballmann and Ledrappier), unfavorable Ricci curvature (Lohkamp), scalar curvature (Delanoë, Hebey and Vaugon), overall curvature (Shioya)], spectrum of the Laplacian (Anné, Colin de Verdière, Matheus, Pesce), isoperimetric and isosystolic inequalities (Calabi, Carron, Gromov), including a few comparable matters [Alexandrov areas (Shiohama and Tanaka, Yamaguchi), elastica (Koiso), subriemannian geometry (Valère and Pelletier)]. Authors are as a rule geometers who labored with Marcel Berger at it slow, and in addition a few more youthful ones. a few papers (Bryant, Colin, Grove...) comprise a short evaluate of contemporary ends up in their specific fields, with the non-experts in brain.

1. time table of the Mathematical talks given on the around Table

Lundi thirteen juillet 1992

K. GROVE : tough and delicate sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems less than crucial speculation for curvature

Mardi 14 juillet 1992

Y. COLIN DE VERDIERE : Spectre et topologie
H. PESCE : Isospectral nilmanifolds
F. MATHEUS : Circle packings and conformal approximation
R. MICHEL : From warmth equation to Hamilton-Jacobi equation
C. ANNE : Formes diff´erentielles sur les vari´et´es avec des anses fines
G. CARRON : In´egalit´e isop´erim´etrique de Faber-Krahn

Mercredi 15 juillet 1992

E. CALABI : in the direction of extremal metrics for isosystolic inequality for closed orientable
surfaces with genus > 1
M. GROMOV : Isosystols
Ch. CROKE : Which Riemannian manifolds are made up our minds through their geodesic flows

Jeudi sixteen juillet 1992

R. BRYANT : Classical, unheard of and unique holonomies : a standing report
T. SHIOYA : habit of maximal geodesics in Riemannian planes
L. VALERE-BOUCHE : Geodesics in subriemannian singular geometry and control
theory
D. GROMOLL : optimistic Ricci curvature : a few fresh developements
Ph. DELANOE : Ni’s thesis revisited
E. HEBEY : From the Yamabe challenge to the equivariant Yamabe problem
Vendredi 17 juillet 1992
W. BALLMANN : Brownian movement, Harmonic features and Martin boundary
U. ABRESCH : Graph manifolds, ends of negatively curved areas and the hyperbolic
120-cell space
N. KOISO : Elastica
Jerry KAZDAN : Why a few differential equations haven't any solutions
J. P. BOURGUIGNON : challenge session

2. at the contributions

Among the above pointed out meetings, 5 are usually not reproduced in those notes,
namely these through Christopher CROKE, Detlef GROMOLL, Jerry KAZDAN, Ren´e
MICHEL and Gilles ROBERT.

Some of them were released in other places, specifically :

CROKE, KLEINER :
Conjugacy and stress for manifolds with a parallel vector field
J. Differential Geom. 39 (1994), 659-680.
LE COUTURIER, ROBERT :
Lp pinching and the geometry of compact Riemannian manifolds
Comment. Math. Helvetici sixty nine (1994), 249-271.
On the opposite hand, Professor SHIOHAMA, who was once invited to provide a conversation, had
not been in a position to come to the desk Ronde. He sought after however to give a
contribution to Marcel Berger. it's been extra to this quantity.

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Extra info for Actes de la Table ronde de geometrie differentielle: En l'honneur de Marcel Berger

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J ξj . The dual bilinear form pj = . , Pj . coincides with (1 + xj )−1 DKj , DKj , and thus Pj and pj are actually real analytic tensor fields on all of Hn . 7) pvj := . , vj vj , . pξj := . , ξj ξj , . Still, these fields constitute a of pj do not even extend continuously across Hn−2 j useful shorthand notation. 7 ) ´ ` 1 SEMINAIRES & CONGRES paj :=pvj − pξj and pbj := . , vj ξj , . + . , ξj vj , . 1. Lemma. 7. 9 ) −2 Tj := η 2 (1 + xj ) 2h (xj ) − x−1 j h (xj ) + xj h(xj ) pj The basic facts about the ∧ –product ∧ pj .

1l − Pi ) . vanish along Hn−2 ∩ UI , and this claim is i ˆ are totally geodesic. 8 (ii). 9. If one wants to drop them, one has to change the construction of the new metric g in such a way that the strata ˆ are automatically totally geodesic. SˆI ⊂ M ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 40 U. ABRESCH V. 3. 5. Lemma. 7. Then, there exist constants c6 and c7 ˆ such that on UI ⊂ Hn depending just on n, h, d0 , and N (i) DGJ\I ≤ c6 η 2 , (ii) D2 GJ\I (iii) DBJ\I ≤ c7 η 2 , 3 ≤ c7 η 2 . 2 Proof.

9 there exist continuous functions cˆ11 , cˆ12 : [0, ∞) → [0, ∞) such that on any domain Ω ∩ UI ⊂ Hn the following estimates hold for any i, i1 , i2 ∈ I with i1 = i2 (i) Pi (1l + GI ) G−1 (1l − Pi ) (ii) Pi1 LI Pi2 1/2 ≤ η 2 cˆ11 (η) xi (1 + xi ) 1/2 1/2 ≤ η 2 cˆ12 (η) xi1 xi2 (1 + xi1 ) −1/2 −1/2 (1 + xi2 ) , −1/2 . 6. However, the symmetry argument used in the proof of that Lemma does not apply directly, since neither (1l + GI ) G−1 nor LI extends even continously from Ω ∩ UI to all of UI . Proof.

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