We consider the evolution of a surface of revolution with boundary Sigma(t) in R-3 by the harmonic mean curvature flow (HMCF) where each point P moves in the normal inward direction with velocity equal to the harmonic mean curvature of the surface. We assume that the principal eigenvalues lambda(1) and lambda(2) of the initial surface have opposite signs, namely K = lambda(1)lambda(2) < 0, while H = lambda(1) + lambda(2) < 0. We show that there exists a time T-0 > 0 for which the (HMCF) admits a unique solution Sigma(t) up to T-0 such that H < 0 for all t < T-0 and (H) over tilde(., T-0) = 0 on some set of sufficiently large measure. In addition, the boundary of the surface evolves by the curve shortening flow.
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