Surfaces of bounded mean curvature in Riemannian manifolds

Details Surfaces of bounded mean curvature in Riemannian manifolds Surfaces of bounded mean curvature in Riemannian manifolds

2008-11-12

arxiv.org/abs/0811.1820v1

Mathematics

Differential Geometry

26 pages

Scientific paper

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 $C^0$ 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 $F\subset M$ 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.

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