MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

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.

geometric evolution, mean curvature flow, maximal regularity

Related publications

2003 Repository Open Access
Nicholas Alikakos and Alexander Freire

The normalized mean curvature flow for a small bubble in a Riemannian manifold

In: Journal of differential geometry, 64 (2003) 2, pp. 247-303