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 l. a. 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 available 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 was once geared up on the CIRM in Luminy (France) in honour of Marcel Berger. In those court cases, contributions disguise many of the fields studied through Marcel Berger in Differential Geometry, particularly : holonomy (Bryant), curvature [positive sectional curvature (Grove), destructive sectional curvature (Abresch and Schroeder, Ballmann and Ledrappier), damaging 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 similar topics [Alexandrov areas (Shiohama and Tanaka, Yamaguchi), elastica (Koiso), subriemannian geometry (Valère and Pelletier)]. Authors are often geometers who labored with Marcel Berger at a while, and in addition a few more youthful ones. a few papers (Bryant, Colin, Grove...) comprise a short overview of modern leads to 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 : difficult and smooth sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems lower than fundamental 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 by way of 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 : confident 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 don't have any solutions
J. P. BOURGUIGNON : challenge session

2. at the contributions

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

Some of them were released somewhere else, specifically :

CROKE, KLEINER :
Conjugacy and pressure 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 used to be invited to offer a conversation, had
not been capable of come to the desk Ronde. He sought after however to offer a
contribution to Marcel Berger. it's been additional to this quantity.

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21) = ∧ (1l+G )−1 I Bi 1+η 2 h (xi ) ϕ0 (η, xi ) − ϕ1 (η, xi ) + 2η 2 (1+xi )h (xi ) pi ∧ pi . 3. 18). Of course, −g0 ∧ g0 is negative definite. 6. Remarks. 4 we may absorb the third, fourth, and nineth term in our expression for R# into −g0 ∧ g0 , provided η > 0 is sufficiently small. 7 implies that c4 := supx≥0 |h (x)| and c5 := supx≥0 (1 + x)|h (x)| are finite numbers. 18) by −g0 ∧ g0 for small positive values of η, too. 22) ΦI := ϕ0 (η, xi ) pi i∈I ´ ` 1 SEMINAIRES & CONGRES ∧ pi 33 ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE which is a sum of manifestly positive semidefinite tensor fields which are defined on the domain Ω ⊂ Hn .

Note that the terms gJ\I and BJ\I are actually real analytic tensor fields on all of UI and not just on Ω ∩ UI . 3. Lemma. — Let I ⊂ J be as above. 8. Then, the following identities hold n ´ ` 1 SEMINAIRES & CONGRES 39 ANALYTIC MANIFOLDS OF NONPOSITIVE CURVATURE along Hn−2 ∩ UI for any i ∈ I i Pi GJ\I (1l − Pi ) = (1l − Pi )GJ\I Pi = 0 (i) Pi BJ\I (1l − Pi ) . , (1l − Pi ) . (ii) Similarly, (1l − Pi )BJ\I Pi . , (1l − Pi ) . = 0. and (1l − Pi )BJ\I (1l − Pi ) . , Pi . Moreover, Pi BJ\I Pi . , Pi .

Corollary. — Let z = z0 + ξ ∈ Cn,1 with ξ , ξ . , KjC |z KjC |z , . C C h ≤ 2 1 + 2a(a + ≤ a2 . Then, h √ 2) 1 + 2xj (z0 ) 2 . Proof. 2 ) shows that X , KjC |z KjC |z , Y 2 = z , e2j C − z , e1j C C e1j , X z , e2j e1j , Y C + z , e1j e1j , X C e2j , Y C C 2 e2j , X C e2j , Y C + e2j , X C e1j , Y C X , e2j C C C . 2 we have eµj , z 2 C C 2 2 z , ej C X , e2j 2 C z , e1j +2 √ 2 + 1 + 2a(a + 2) eµj , z0 √ 1 ≤ 1 + 2a(a + 2) 1 + 2xj (z0 ) . 6) yield eµj , X 2 C ≤ 1 + 2 eµj , z0 2 X ,X ≤ 1 + 2xj (z0 ) X , X h .

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