The three geodesics theorem of Lusternik and Schnirelmann asserts that for every Riemannian metric on the 2-sphere, there exist at least three embedded closed geodesics. In the process of determining the geodesics as critical points of the energy or length functional, a suitable method of curve shortening is needed. It has been suggested to use the so-called curve shortening flow as it continuously deforms smooth embedded curves while naturally preserving their embeddedness. In the 1980s, the investigation of the curve shortening flow began and a proof of the Lusternik-Schnirelmann theorem using the flow was sketched. We build upon these results. After introducing the curve shortening flow, we prove the well-known result that the geodesic curvature of a smooth embedded closed curve on a smooth closed two-dimensional Riemannian manifold decreases smoothly to zero, provided the curve evolves forever under the flow. From this, we prove subconvergence to an embedded closed geodesic, using mainly local arguments. After introducing, in the form of Lusternik-Schnirelmann theory, the topological machinery employed in the process of determining critical points of certain functions, we turn to the three geodesics theorem which we prove under a few assumptions. For the round metric on the 2-sphere, we deformation retract a suitable space of unparametrized curves onto a simpler space of which we determine the homology groups relative to a subspace which deformation retracts onto the subspace of point curves. As this yields three subordinate homology classes, proving the validity of Lusternik-Schnirelmann theory for the curve shortening flow and the length functional on our space of curves completes the proof.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:16857 |
| Date | 05 December 2017 |
| Creators | Sewerin, Sebastian |
| Contributors | Ziller, Wolfgang, Rademacher, Hans-Bert, Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English, German |
| Detected Language | English |
| Type | info:eu-repo/semantics/updatedVersion, doc-type:masterThesis, info:eu-repo/semantics/masterThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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