Abstract
We consider a compact star-shaped mean convex hypersurface \({\Sigma^2\subset \mathbb{R}^3}\). We prove that in some cases the flow exists until it shrinks to a point. We also prove that in the case of a surface of revolution which is star-shaped and mean convex, a smooth solution always exists up to some finite time T < ∞ at which the flow shrinks to a point asymptotically spherically.
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Communicated by G. Huisken.
P. Daskalopoulos was partially supported by NSF Grants 0701045 and 0354639 and N. Sesum was partially supported by NSF Grant 0604657.
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Daskalopoulos, P., Sesum, N. The harmonic mean curvature flow of nonconvex surfaces in \({\mathbb{R}^3}\) . Calc. Var. 37, 187–215 (2010). https://doi.org/10.1007/s00526-009-0258-x
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DOI: https://doi.org/10.1007/s00526-009-0258-x