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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.

The harmonic mean curvature flow of nonconvex surfaces in $${\mathbb{R}^3}$$