Curvature flows on surfaces with boundary

Abstract

A theorem of R.Hamilton [6] and B.Chow states that for every initial metric the initial value problem (1) possesses a unique solution which is defined for all t ≥ 0. Moreover, for t → ∞ the solution converges exponentially to a metric of constant Gauss curvature. Different proofs for this result were given by J.Bartz, M.Struwe and R.Ye [2] and by M.Struwe [13]. On a surface with boundary, one can demand that the Gauss curvature in the interior is constant and the geodesic curvature at the boundary vanishes. Alternatively, one can require that the geodesic curvature at the boundary is constant and the Gauss curvature in the interior vanishes. The evolution equation corresponding to the first case is