Wintersemester 2010/11
Location: Mathematisches Institut, Universität zu Köln
Freitag 3.12.10 |
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. |
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Samstag 4.12.10 |
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 R^{2n}. |
This lecture series is supported by the Graduiertenkolleg
"Globale Strukturen in Geometrie und Analysis".
H. Geiges, 18.11.10