Surfaces of Bounded Mean Curvature in Riemannian Manifolds

Consider a sequence of closed, orientable surfaces of fixed genus g in a Riemannian manifold M with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the inclusion maps converge in C to a map from a surface of genus g to M . We also show that, on passing to a further subsequence, the distance functions corresponding to pullback metrics converge to a pseudo-metric of fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface F ⊂ M together with bounds on the geometry of M give an upper bound on the diameter of F . Our proof is modelled on Gromov’s compactness theorem for J-holomorphic curves.