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The Normalized Mean Curvature Flow For A Small Bubble In A Riemannian Manifold
Nicholas Alikakos and Alexander Freire
We study the effect of the curvature of the ambient space on the evolution of small, almost geodesic spheres ("bubbles")under the normalized mean curvature flow . We establish the robustness of the almost spherical shape globally in time and show that the center of mass of the bubble moves, to principal order,by the gradient of the scalar curvature. We comment that we expect this law of motion of the center of mass to hold,modulo a coefficient, for a large class of perimeter shortening, volume preserving flows.