Shortest Curves Associated to a Degenerate Jacobi Metric on the two Torus

Shortest Curves Associated to a Degenerate Jacobi Metric on the two Torus

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John Mather, Princeton University
Fine Hall 401

Let $P$ be a potential on the two torus that takes its minimum value at a unique point m. Set $E_0 := P(m)$. For a real number $E$, let $g_E$ be the Jacobi metric associated to $P$ and $E$. For $E \gt E_0, g_E$ is a Riemannian metric. An ancient theorem of Morse and Hedlund says that a $g_E$-shortest curve in an indivisible homology class is simple. For $E = E_0, g_E$ is no longer a Riemannian metric because it vanishes at $m$. (It is a Riemannian metric in the complement of $m$.) For a suitable potential $P$, and a suitable indivisible homology class $h$, a $g_{E_0}$-shortest curve in $h$ crosses itself at $m$, so the theorem of Hedlund and Morse does not generalize to the case $E = E_0$. In this talk, I will describe examples of such shortest curves that cross themselves and give a few ideas of how to prove that each such shortest curve does not cross itself except at m and is a bouquet of simple curves in at most three homology classes.