Curve shortening and the topology of closed geodesics on surfaces. (English) Zbl 1137.53330

Angenent, Sigurd B.

Ann. Math. (2) 162, No. 3, 1187-1241 (2005).

Summary: 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.

MSC:

53C22	Geodesics in global differential geometry
58D10	Spaces of embeddings and immersions
53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

flat knot types; compact surfaces; Conley index