In 1976, the author (see the open problem section in [3]) asked the following question: if p is a function defined on i? 3, what is the condition on p so that there is a sphere in R3 whose mean curvature is given by p. There were important works by Treibergs-Wei [2] and others on this question. Recently in a lecture that I gave in the Chinese University of Hong Kong, I realized that my work with R. Schoen on the existence of black hole can be used to settle the prescribed mean curvature question for a large class of p.

In this paper we prove a monotonicity formula for the integral of the mean curvature for complete and proper hypersurfaces of the hyperbolic space and, as consequences, we obtain a lower bound for the integral of the mean curvature and that the integral of the mean curvature is infinity.

A generation of symbols asserted for n ≥ 0 in the proof of Theorem 3.3 of the original paper in fact only holds for n > 0, thus undermining the proof of the theorem. A new version of Section 3.5 of the original paper is given, culminating in a corrected proof of Theorem 3.3. The author thanks Deepak Khosla for pointing out the gap in the previous version of the proof.

We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.

Let fn → f0 be a convergent sequence of rational maps, preserving critical relations, and f0 be geometrically finite with parabolic points. It is known that for some unlucky choices of sequences fn,the Julia sets J(fn) and their Hausdorff dimensions may fail to converge as n → ∞. Our main result here is to prove the convergence of J(fn) and H.dim J(fn) for generic sequences fn. The same conclusion was obtained earlier, with stronger hypotheses on the sequence fn, by Bodart-Zinsmeister and then by McMullen. We characterize those choices of fn by means of flows of appropriate polynomial vector fields (following Douady-Estrada-Sentenac). We first prove an independent result about the (s-dimensional) length of separatrices of such flows, and then use it to estimate tails of Poincar´e series. This, together with existing techniques,provides the desired control of conformal densities and Hausdorff dimensions. Our method may be applied to other problems relatedto parabolic perturbations.