The Bergman metric on a Stein manifold with a bounded plurisubharmonic function

In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature , which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.     

Global existence from single-component estimates in a semilinear reaction-diffusion system

For a system of two reaction-diffusion equations coupled by power nonlinearities, we prove that an bound on a single component for suitable is enough to guarantee global existence. Also we provide a strong indication that our condition on is the best possible. Moreover, this continuation result is in contrast with the corresponding necessary and sufficient conditions for local existence obtained earlier by the authors.

     

Differentiable mappings with an infinite number of critical points

In this paper we shall give some sufficient conditions in order that the so-called -category of a pair of differentiable manifolds be infinite.

     

Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with

Boas and Straube proved a general sufficient condition for global regularity of the -Neumann problem in terms of families of vector fields that commute approximately with . In this paper, we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds. This question turns out to be closely related to properties of interest from the point of view of foliation theory.

     

Hyperbolic rank of products

Generalizing a result of Brady and Farb (1998), we prove the existence of a bilipschitz embedded manifold of negative curvature bounded away from zero and dimension in the product of two Hadamard manifolds of dimension with negative curvature bounded away from zero.

Combining this result with a result of Buyalo and Schroeder (2002), we prove the additivity of the hyperbolic rank for products of manifolds with negative curvature bounded away from zero.

     

Metrics of positive curvature with conic singularities on the sphere

A simple proof is given of the necessary and sufficient condition on a triple of positive numbers for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles . The same condition is necessary and sufficient for the triple to be interior angles of a spherical triangular membrane.

     

On the sum of the index of a parabolic subalgebra and of its nilpotent radical

In this short note, we investigate the following question of Panyushev stated in 2003: ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra and the index of its nilpotent radical always greater than or equal to the rank of ?' Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author and proved by Fauquant-Millet and Joseph in 2005 and Joseph in 2006, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of . This provides new examples of direct sum decomposition of a semisimple Lie algebra verifying the ``index additivity condition' as stated by Raïs.

     

Conformally invariant powers of the Laplacian -- A complete nonexistence theorem

We show that on conformal manifolds of even dimension there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian for n/2$">. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of for , is sharp.

     

Hardy spaces and partial derivatives of conjugate harmonic functions

In this paper we give necessary and sufficient conditions for a harmonic vector and all its partial derivatives to belong to for all .

     

Admissible metrics in the -Yamabe equation

In most previous works on the existence of solutions to the -Yamabe problem, one assumes that the initial metric is -admissible. This is a pointwise condition. In this paper we prove that this condition can be replaced by a weaker integral condition.

     

Stochastic processes with sample paths in reproducing kernel Hilbert spaces

A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either or . Driscoll also found a necessary and sufficient condition for that probability to be .

Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem - specifically the necessity of the condition - turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available.

Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.

     

Complete manifolds with nonnegative curvature operator

In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with -nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension ) whose curvature operator is bounded and satisfies the pinching condition , for some , must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.

     

A note on finite abelian gerbes over toric Deligne-Mumford stacks

Any toric Deligne-Mumford stack is a -gerbe over the underlying toric orbifold for a finite abelian group . In this paper we give a sufficient condition so that certain kinds of gerbes over a toric Deligne-Mumford stack are again toric Deligne-Mumford stacks.

     

Only `free' measures are admissable on is incomplete

Using elementary arguments and without having to recall the Gleason Theorem, we prove that the existence of a nonsingular measure on the lattice of orthogonally closed subspaces of an inner product space is a sufficient (and of course, a necessary) condition for to be a Hilbert space.

     

Random variable dilation equation and multidimensional prescale functions

A random variable satisfying the random variable dilation equation , where is a discrete random variable independent of with values in a lattice and weights and is an expanding and -preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density which will satisfy a dilation equation

We have obtained necessary and sufficient conditions for the existence of the density and a simple sufficient condition for 's existence in terms of the weights Wavelets in can be generated in several ways. One is through a multiresolution analysis of generated by a compactly supported prescale function . The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when is a prescale function.

     

Supercongruences between truncated hypergeometric functions and their Gaussian analogs

Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension . For manifolds of dimension , he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.

     

Composition operators on Hardy spaces on Lavrentiev domains

For any simply connected domain , we prove that a Littlewood type inequality is necessary for boundedness of composition operators on , , whenever the symbols are finitely-valent. Moreover, the corresponding ``little-oh' condition is also necessary for the compactness. Nevertheless, it is shown that such an inequality is not sufficient for characterizing bounded composition operators even induced by univalent symbols. Furthermore, such inequality is no longer necessary if we drop the extra assumption on the symbol of being finitely-valent. In particular, this solves a question posed by Shapiro and Smith (2003). Finally, we show a striking link between the geometry of the underlying domain and the symbol inducing the composition operator in , and in this sense, we relate both facts characterizing bounded and compact composition operators whenever is a Lavrentiev domain.

     

Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature

We give a sufficient condition for a higher dimensional Kleinian group to be convex cocompact in terms of the critical exponent of . As a consequence, we see that the fundamental group of a compact conformally flat manifold with positive scalar curvature is hyperbolic in the sense of Gromov. We give some other applications to geometry and topology of conformally flat manifolds with positive scalar curvature.

     

The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature

In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampère equations in a strictly convex domain to an arbitrary smooth bounded domain in as well as in a general Riemannian manifold. We prove for the nondegenerate case that a sufficient (and necessary) condition for the classical solvability is the existence of a subsolution. For the totally degenerate case we show that the solution is in if the given boundary data extends to a locally strictly convex function on . As an application we prove some existence results for spacelike hypersurfaces of constant Gauss-Kronecker curvature in Minkowski space spanning a prescribed boundary.

     

Uniformly bounded maximal -disks, Bers space and harmonic maps

We characterize Bers space by means of maximal -disks. As an application we show that the Hopf differential of a quasiregular harmonic map with respect to strongly negatively curved metric belongs to Bers space. Also we give further sufficient or necessary conditions for a holomorphic function to belong to Bers space.