UBC Theses and Dissertations
Length-minimizing closed curves on manifolds with boundary Kohut, Hannah
The purpose of this thesis is to explore the properties of closed curves which are length-minimizing in a nontrivial homotopy class of a compact Riemannian manifold with boundary. This project was inspired by a question asked by Professor Liam Watson about the existence of length-minimizing curves on a torus with finitely many disks removed, and whether these curves leave the boundary components tangentially. In our research, we extend the question to compact smooth Riemannian manifolds with smooth boundary. We first discuss some preliminary results to determine an appropriate class of curves to minimize over. We then explore the properties of the length and energy functionals, showing that a minimizer of the energy is also a minimizer of the length. We directly minimize the energy functional to show the existence of a length-minimizing curve in any nontrivial homotopy class of a compact Riemannian manifold with boundary. Finally, we address the regularity of length-minimizing curves, showing that they are piecewise geodesics (possibly with infinitely many pieces).
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