The heat flow for subharmonic orbits (Q1892327)

This paper investigates the heat flow for subharmonic orbits on a compact Riemannian manifold \(M\). If \(V : M \to \mathbb{R}\) is a smooth potential function, it gives rise to a second order Hamiltonian system. If the action functional \[ J(\gamma (t)) = \int{\textstyle{1\over 2}}\langle \gamma', \gamma'\rangle - V(\gamma (t)) \] is a Morse function, the author proves that the heat flow for subharmonics exists globally and converges to a critical point of the energy. (A subharmonic orbit is a critical point of the action functional.) As a corollary, the author's result establishes convergence of the geodesic heat flow to a geodesic without any assumptions about the curvature of \(M\).