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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), l. a. courbure [courbure sectionnelle confident (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 complaints, contributions disguise lots of the fields studied by means of Marcel Berger in Differential Geometry, particularly : holonomy (Bryant), curvature [positive sectional curvature (Grove), detrimental sectional curvature (Abresch and Schroeder, Ballmann and Ledrappier), destructive 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 topics [Alexandrov areas (Shiohama and Tanaka, Yamaguchi), elastica (Koiso), subriemannian geometry (Valère and Pelletier)]. Authors are commonly geometers who labored with Marcel Berger at it slow, and likewise a few more youthful ones. a few papers (Bryant, Colin, Grove...) contain a quick assessment of contemporary leads to their specific fields, with the non-experts in brain.

1. agenda of the Mathematical talks given on the around Table

Lundi thirteen juillet 1992

K. GROVE : challenging and gentle 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 via their geodesic flows

Jeudi sixteen juillet 1992

R. BRYANT : Classical, extraordinary 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 contemporary 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 capabilities 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 aren't 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 different places, particularly :

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

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

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1, we need to establish uniform convergence of C the series gJ\I = j∈J\I gjC in an arbitrarily small open neighborhood in Cn,1 of any given point z0 ∈ UI . Here, we may work with respect to any norm on Cn,1 which may even depend on z0 . By homogeneity, we can assume that z0 = (0, . . , 0, 1),6 and as a norm we take the Hermitian inner product , h from the standard identification of Cn,1 with Cn × C. 2. Lemma. — Let z = z0 + ξ ∈ Cn,1 with ξ , ξ . h ≤ a2 . Then, for any e ∈ Rn,1 h with e , e = 1, the following inequalities hold (i) e,z 2 C− 2 C e , z0 xC j (z) − xj (z0 ) (ii) ≤a a+ √ 2 2 1 + 2 e , z0 C , √ ≤ 2a a + 2 1 + xj (z0 ) .

In fact, the expression for R# shows directly5 that for sufficiently small values of ˆi over the stratum Si of the divisor which is η the sectional curvature of any plane E spanned by the unit normal vector of Sˆi and the tangent vector of the fibration Sˆi → Si is approximately −η −2 . Moreover, the region where the sectional curvature gets large in absolute value concentrates more and more along the preimage of the divisor. This behaviour is best understood when considering the Gauß–Bonnet Theorem, figuring out what it means to add a cross–cap of size ∼ η to a fixed ball orthogonal to Si ⊂ Hn−2 .

1. (iii) A non–trivial vector w ∈ Hpˆ, pˆ ∈ SˆI , is orthogonal to νpˆSˆI with respect to ˆ if and only if its image dπI |pˆ w ∈ Tπ (ˆp) Hn is orthogonal to all the metric g on M I spaces Ei|π I (p) ˆ , i ∈ I, with respect to the hyperbolic metric. 3). ˆ Proof. 1). By continuity they ˆˆ I . 7) gJ\I dπI vi , dπI vi 1 = 0 2 for any pair of distinct indices i1 , i2 ∈ I . 1. 7 ).

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