Let H be a semisimple algebraic group. We prove the semistable reduction theorem for μ–semistable principal H–bundles over a smooth projective variety X defined over the field C. When X is a smooth projective surface and H is simple, we construct the algebro–geometric Donaldson–Uhlenbeck compactification of the moduli space of μ–semistable principal H–bundles with fixed char- acteristic classes and describe its points. For large characteristic classes we show that the moduli space of μ–stable principal H–bundles is non–empty.

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.

We study the question of local and global uniqueness of completions, based on null geodesics, of Lorentzian manifolds. We show local uniqueness of such boundary extensions. We give a necessary and sufficient condition for existence of unique maximal completions. The condition is verified in several situations of interest. This leads to existence and uniqueness of maximal spacelike conformal boundaries, of maximal strongly causal boundaries, as well as uniqueness of conformal boundary extensions for asymptotically simple space-times. Examples of applications include the definition of mass, or the classification of inequivalent extensions across a Cauchy horizon of the Taub space-time.

In this paper, we first obtain the sub-Laplacian comparison theorem in a complete noncompact pseudohermitian manifold of vanishing torsion (i.e. Sasakian manifold). Secondly, we derive the sub-gradient estimate for positive pseudoharmonic functions in a complete noncompact pseudohermitian manifold which satisfies the CR sub-Laplacian comparison property. It is served as the CR analogue of Yauís gradient estimate. As a consequence, we have the natural CR analogue of Liouville-type theorems in a complete noncompact Sasakian manifold of nonnegative pseudohermitian Ricci curvature tensors.

The CD inequalities are introduced to imply the gradient estimate of Laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,1) and CD(K,n).