Mathematical Research Letters

Volume 29 (2022)

Number 4

Inverse mean curvature flow over non-star-shaped surfaces

Pages: 1065 – 1086



Brian Harvie (Department of Mathematics, University of California, Davis, Calif., U.S.A.)


We derive an upper bound on the waiting time for a variational weak solution to Inverse Mean Curvature Flow in $\mathbb{R}^{n+1}$ to become star-shaped. As a consequence, we demonstrate that any connected surface moving by the flow which is not initially a topological sphere develops a singularity or self-intersection within a prescribed time interval depending only on initial data. Finally, we establish the existence of either finite-time singularities or intersections for certain topological spheres under IMCF.

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Received 6 February 2020

Accepted 18 August 2020

Published 23 February 2023