Lecture series on Isosystolic Inequalities

Prof. Juan Carlos Álvarez Paiva (Lille)

Wintersemester 2010/11

Location: Mathematisches Institut, Universität zu Köln

Can Humpty-Dumpty increase his waistline without distending his skin?
Seminarraum 1, 10:30-11:30

Abstract: In this talk we shall explore the relation between the volume of a compact Riemannian n-dimensional manifold (M,g) and the length of its shortest closed geodesic. For this purpose we define the systolic volume S(M,g) as the volume of (M,g) divided by the n-th power of the length of its shortest closed geodesic. Unfortunately, almost nothing is known about this quantity - at least when M is simply connected - beyond the following result:

Theorem (Croke, Rotman). The systolic volume of any Riemannian metric on the 2-sphere is greater than 1/32.

It is thought that compact rank-one symmetric spaces are local minima for the systolic volume. In this talk I shall show how some simple contact geometry and bifurcation theory yields that they are at least critical points of the systolic volume considered as a functional on the space of Riemannian metrics.

Definition. The metric g is a critical point of the systolic volume if for any smooth deformation g(t) of g the limit as t tends to zero of the increment quotient (S(M,g(t)) - S(M,g(0)))/t is equal to zero whenever it exists.

Theorem (Alvarez-Balacheff). If all geodesics of (M,g) are closed and of the same length, then (M,g) is a critical point of the systolic volume. Moreover, if we relax the variational problem and allow deformations by smooth Finsler metrics, then these are the only critical points.
Isosystolic inequalities in contact geometry and a question of Viterbo
Hörsaal, 14:15-15:15

Abstract: After reformulating systolic geometry in contact-geometric terms, we will see that a result similar to that formulated in the first lecture holds for capacity-volume inequalities of convex hypersurfaces in R2n.

This lecture series is supported by the Graduiertenkolleg "Globale Strukturen in Geometrie und Analysis".

H. Geiges, 18.11.10