tailieunhanh - Đề tài " Curve shortening and the topology of closed geodesics on surfaces "

We study “flat knot types” of geodesics on compact surfaces M 2 . For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M 2 . We conclude existence of closed geodesics with prescribed flat knot types, provided the associated Conley index is nontrivial. 1. Introduction If M is a surface with a Riemannian metric g then closed geodesics on (M, g) are critical points of the length functional L(γ) = |γ (x)|dx defined on the space of unparametrized C. | Annals of Mathematics Curve shortening and the topology of closed geodesics on surfaces By Sigurd B. Angenent Annals of Mathematics 162 2005 1187-1241 Curve shortening and the topology of closed geodesics on surfaces By Sigurd B. Angenent Abstract We study flat knot types of geodesics on compact surfaces M2. For every flat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening flow on the space of immersed curves on M2. We conclude existence of closed geodesics with prescribed flat knot types provided the associated Conley index is nontrivial. 1. Introduction If M is a surface with a Riemannian metric g then closed geodesics on M g are critical points of the length functional L y J 7z d defined on the space of unparametrized C2 immersed curves with orientation . we consider closed geodesics to be elements of the space Q Imm S1 M Diff S1 . Here Imm S 1 M 7 G C2 S 1 M 7 0 for all e G S1 and Diff S1 is the group of C2 orientation preserving diffeomorphisms of S1 R Z. We will abuse notation freely and use the same symbol 7 to denote both a convenient parametrization in C2 S1 M and its corresponding equivalence class in Q. The natural gradient flow of the length functional is given by curve shortening . by the evolution equation 1 ỡ Ỡ27 T def d7 dt ds2 T dS In 1905 Poincare 33 pointed out that geodesics on surfaces are immersed curves without self-tangencies. Similarly different geodesics cannot be tangent - all their intersections must be transverse. This allows one to classify closed geodesics by their number of self-intersections or their flat knot type Supported by NSF through a grant from DMS and by the NWO through grant NWO-600-61-410. 1188 SIGURD B. ANGENENT and to ask how many closed geodesics of a given type exist on a given surface M g . Our main observation here is that the curve shortening flow 1 is the right tool to deal with this question. We formalize these notions in the following definitions which .