Complete algebraic vector fields on affine surfaces-Part II

In this work we finish off the classification of meromorphic semi-complete vector fields announced in Rebelo [J. Geom. Anal 13(4) (2003) 669-696]. As an application of our results, we give a simple and more geometric proof of the classification of complete polynomial vector fields on C² recently obtained through the works of Brunella and McQuillan.     

Toral algebraic sets and function theory on polydisks

A toral algebraic set A is an algebraic set in ℂ ⁿ whose intersection with T ⁿ is sufficiently large to determine the holomorphic functions on A. We develop the theory of these sets, and give a number of applications to function theory in several variables and operator theoretic model theory. In particular, we show that the uniqueness set for an extremal Pick problem on the bidisk is a toral algebraic set, that rational inner functions have zero sets whose irreducible components are not toral, and that the model theory for a commuting pair of contractions with finite defect lives naturally on a toral algebraic set.     

A generalized affine isoperimetric inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.     

Maximum boundary regularity of bounded Hua-harmonic functions on tube domains

In this article we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of the Heisenberg group or equivalently on the Siegel upper half-plane.     

On the nonextendability of germs of holomorphic functions

We construct explicit supporting manifolds and local holomorphic peak functions as obstructions to the extendability of holomorphic functions on a class of domains not necessarily pseudoconvex in C^N, N >2.     

Fibered neighborhoods of curves in surfaces

The aim of this article is to study fibered neighborhoods of compact holomorphic curves embedded in surfaces. It is shown that when the self-intersection number of the curve is sufficiently negative the fibration is equivalent to the linear one defined in the normal bundle to the curve. The obstructions to equivalence in the general case are described.     

On the stability of the behavior of random walks on groups

We show that, for random walks on Cayley graphs, the long time behavior of the probability of return after 2n steps is invariant by quasi-isometry.     

On Hamiltonian stable minimal Lagrangian surfaces in CP2

We study Hamiltonian stable minimal Lagrangian closed submanifolds in the standard complex projective n-space CP ⁿ .It is shown that when n = 2such a surface Σis either totally geodesic or flat if the multiplicity of the Laplacian acting on C∞(Σ)is less than or equal to 6.     

Regularity of area-minimizing surfaces in 3D polytopes and of invariant hypersurfaces inR n

In (the surface of) a convex polytope P ³ inR ⁴,an areaminimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an areaminimizing G-invariant oriented hypersurface is smooth (except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also treat other categories of surfaces such as rectifiable currents modulo v and soap films.     

Growth properties of plurisubharmonic functions related to Fourier-Laplace transforms

The purpose of the paper is to study the behavior at infinity of Fourier-Laplace transforms of distributions or more generally plurisubharmonic functions u in Cⁿ with bounds of the form

Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group

We present a characterization of the open unit ball in a separable infinite dimensional Hilbert space by the property of automorphism orbits among the domains that are not necessarily bounded. This generalizes the recent work of Kim and Krantz [6]. Key new features of this article include: a lower bound estimate of the Kobayashi metric and distance near a pluri-subharmonic peak boundary point of the domains in Banach spaces, an effective localization argument, and an improvement of weak-type convergence of sequences of biholomorphic mappings of domains in Banach spaces.     

Intrinsic capacities on compact Kähler manifolds

We study fine properties of quasiplurisubharmonic functions on compact Kähler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally pluripolar sets are globally “quasi-pluripolar.”     

Compact Heisenberg manifolds as CR manifolds

Let M be the quotient of the Heisenberg group by a discrete co-compact subgroup, with the natural strongly pseudoconvex CR structure. We identify the eigenvalues and eigenforms of the Kohn Laplacians on M and show how to realize M as the boundary of a bounded domain in a line bundle over an Abelian variety.     

Defect and evaluations

Let S be a generic submanifold of C ^N of real codimension m. In this paper we continue the study, carried over by various authors, of the set of analytic discs attached to S. Moreover, we look at the subspaces of C ^N obtained by evaluating at given points, holomorphic maps which are infinitesimal deformations of analytic discs attached to S.     

Singular integral operators on Riemann surfaces and elliptic differential systems

The derivatives of the Cauchy kernels on compact Riemann surfaces generate singular integral operators analogous to the Calderón-Zigmund operators with the kernel (t - z)² on the complex plane. These operators play an important role in studying elliptic differential equations, boundary value problems, etc. We consider here the most important case of the multi-valued Cauchy kernel with real normalization of periods. In the opposite plane case, such an operator is not unitary. Nevertheless, its norm in L₂ is equal to one. This result is used to study multi-valued solutions of elliptic differential systems.     

Riesz sequences of translates and generalized duals with support on [0, 1]

If the integer translates of a function ø with compact support generate a frame for a subspace W of L ²(?),then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis belonging to W. Considerable freedom can be obtained by considering oblique duals, i.e., duals not necessarily belonging to W. Extending work by Ben-Artzi and Ron, we characterize the existence of oblique duals generated by a function with support on an interval of length one. If such a generator exists, we show that it can be chosen with desired smoothness. Regardless whether ø is polynomial or not, the same condition implies that a polynomial dual supported on an interval of length one exists.     

Convergence of Rothe's method for fully nonlinear parabolic equations

Convergence of Rothe's method for the fully nonlinear parabolic equation u_t+F(D²u, Du, u, x, t)=0 is considered under some continuity assumptions on F. We show that the Rothe solutions are Lipschitz in time, Hölder in space, and they solve the equation in the viscosity sense. As an immediate corollary we get Lipschitz behavior in time of the viscosity solutions of our equation.     

Holomorphic approximation on compact pseudoconvex complex manifolds

Let be a smoothly bounded compact pseudoconvex complex manifold of finite type in the sense of D’Angelo such that the complex structure of M extends smoothly up to bM. Let m be an arbitrary nonnegative integer. Let f be a function in H(M)∩ H_m(M), where H_m(M) is the Sobolev space of order m. Then f can be approximated by holomorphic functions on in the Sobolev space H_m(M). Also, we get a holomorphic approximation theorem near a boundary point of finite type.