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 l. a. plupart des sujets abordés par Marcel Berger en Géométrie Différentielle et plus précisément : l'holonomie (Bryant), los angeles 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 prepared on the CIRM in Luminy (France) in honour of Marcel Berger. In those complaints, contributions conceal lots of the fields studied by means of Marcel Berger in Differential Geometry, particularly : holonomy (Bryant), curvature [positive sectional curvature (Grove), damaging 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 often geometers who labored with Marcel Berger at your time, and likewise a few more youthful ones. a few papers (Bryant, Colin, Grove...) contain a short evaluate of modern 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 : tough and tender sphere theorems
T. YAMAGUCHI : A convergence theorem for Alexandrov spaces
J. LOKHAMP : Curvature h-principles
G. ROBERT : Pinching theorems lower 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 decided by means of 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 services 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 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 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 resources for Actes de la Table ronde de geometrie differentielle: En l'honneur de Marcel Berger

Example text

11) 2 DX,W g(Y, Z) −2 h (xj ) − 4x−1 4η 2 x−2 j j h (xj ) + 6xj h(xj ) = j∈J Kj , DX Kj Kj , DW Kj Kj , Y Kj , Z 2η 2 x−2 h (xj )−2x−1 j j h(xj ) + j∈J Kj , Y Kj , Z DX Kj , DW Kj − R0 (Kj , X)W , Kj + Kj , Y Kj , DW Kj DX Kj , Z + Kj , DX Kj DW Kj , Z + Kj , Z Kj , DX Kj DW Kj , Y + Kj , DW Kj DX Kj , Y η 2 x−2 j h(xj ) + j∈J DW Kj , Y DX Kj , Z + DX Kj , Y − R0 (Kj , X)W , Y Kj , Z − Kj , Y DW Kj , Z R0 (Kj , X)W , Z . 2 In this computation we have used the identity DX,W Kj + R0 (Kj , X)W = 0 in order to determine the second derivatives D2 Kj .

Remark. 10 that supp(h) ⊂ [0, sinh2 1 2 d0 ]. 18 ) − 2η 2 h (xi ) g0 i∈I − ∧ pξi − 2η 2 (1 + xi ) h (xi ) pi ∧ pi i∈I 4 η h (xi1 ) h (xi2 ) pξi1 ∧ pξi2 . 8 at all. 4. ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 1996 6. 5. 1. Proposition. 9. Then, ˆ ε such that for 0 < η < η2 for any ε > 0, there exists a constant η2 = η2 n, h, d0 , N, the following estimates hold on each domain Ω ∩ UI (i) −ε g0 (ii) −ε g0 ∧ ∧ g 0 + ΦI g 0 + ΦI ≤ ≤ BI ∧ G−1 −(1l+GI )−1 BI BJ\I ∧ G−1 BI ≤ ε g 0 ∧ g 0 + ΦI ≤ ε g 0 ∧ g 0 + ΦI .

We get πI∗ (g0 + gI )|(p,ϕ) = g0 |Tp W U ×Tp W U + I xi + η 2 h(xi ) I |p dϕi 2 . i∈I Evidently, the right hand side describes a real analytic, Stab I -invariant, Riemannian #I metric on all of WIU × (R/2πZ) . (iii) Note that p ∈ SI is contained in some domain UI with I ⊂ I ⊂ J. 5) it is clear that πI−1 {p} is a totally geodesic product torus in WIU × (R/2πZ) #I equipped with the metric πI∗ g0 + gI . If η is sufficiently small, then the function x → x + η 2 h(x), x ≥ 0, takes its absolute minimum precisely at x = 0.

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