How to relate the mean curvature vector evolution over a submanifold of an euclidean space to growth of the volumes of geodesic balls. Can i determine the volume of a geodesic ball by integrating the norm of the mean curvature vector? Thanks in advance
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What do you mean by "growth of the volumes of geodesic balls."? Are you talking about the time evolution of a family of submanifolds, or a fixed submanifold with an intrinsic geodesic ball of increasing radius? – treble Oct 02 '13 at 17:58
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Yes, the second option: a fixed submanifold of euclidean space. How would the evolution of the mean curvature affect the volumes of intrinsic geodesic balls if the radius is being increased. – stirlitz Oct 02 '13 at 19:36