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.     

Characterization of the unit ball in C<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> among complex manifolds of dimension<Emphasis Type="BoldItalic">n</Emphasis>

We show that if the group of holomorphic automorphisms of a connected complex manifold M of dimension n is isomorphic as a topological group equipped with the compact-open topology to the automorphism group of the unit ball B ⁿ ⊂ ℂ ⁿ ,then M is biholomorphically equivalent to B ⁿ.     

Hyperbolic manifolds of dimensionn with automorphism group of dimensionn 2 − 1

We consider complex Kobayashi-hyperbolic manifolds of dimension n ≥ 2 for which the dimension of the group ofholomorphic automorphisms is equal to n² − 1. We give a complete classification of such manifolds for n ≥ 3 and discuss several examples for n = 2.     

Integrability of rough almost complex structures

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.     

Rank two fundamental groups of positively curved manifolds

This paper deals with the construction of previously unknown fundamental groups for positively curved manifolds.     

Infinite-dimensionality of the automorphism groups of homogeneous Stein manifolds

We show that the group of holomorphic automorphisms of a Stein manifold X with dim X ≥ 2 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.     

Stein domains in complex surfaces

Let S be a closed connected real surface and π: S→X a smooth embedding or immersion of S into a complex surface X. We denote by I(π) (resp. by I_±(π) if S is oriented) the number of complex points of π (S)∪X counted with algebraic multiplicities. Assuming that I(π)≤0 (resp. I_±(π)≤0 if S is oriented) we prove that π can be C⁰ approximated by an isotopic immersion π₁: S→X whose image has a basis of open Stein neighborhood in X which are homotopy equivalents to π₁ (S). We obtain precise results for surfaces in and find an immersed symplectic sphere in with a Stein neighborhood.     

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ ^N are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂ ^N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.     

Small eigenvalues of geometrically finite manifolds

Let M be a complete geometrically finite manifold of bounded negative curvature, infinite volume, and dimension at least 3.We give both a lower bound for the bottom of the spectrum of M and an upper bound for the number of the small eigenvalues of M. These bounds only depend on the dimension, curvature bounds and the volume of the oneneighborhood of the convex core.     

The fundamental solution on manifolds with time-dependent metrics

In this article we prove the existence of a fundamental solution for the linear parabolic operator L(u) = (Δ − ∂/∂t − h)u, on a compact n-dimensional manifold M with a time-parameterized family of smooth Riemannian metrics g(t). Δ is the time-dependent Laplacian based on g(t), and h(x, t) is smooth. Uniqueness, positivity, the adjoint property, and the semigroup property hold. We further derive a Harnack inequality for positive solutions of L(u) = 0 on (M, g(t) on a time interval depending on curvature bounds and the dimension of M, and in the special case of Ricci flow, use it to find lower bounds on the fundamental solution of the heat operator in terms of geometric data and an explicit Euclidean type heat kernel.     

On strongly pseudoconvex manifolds with one-dimensional exceptional set

Let X be a strongly pseudoconvex n-dimensional manifold with one-dimensional connected exceptional set S. Here we show that if S is reducible, then X is embeddable into C^N ×P ^{m for some N, m and in particular X is Kählerian with a possible exception for n = 3; we analyze this exceptional case but we do not know if it may occur. The case in which S is irreducible was previously analyzed by Tan [11], who proved that X is embeddable if either n ≠ 3 or S is not isomorphic to} P ¹. In the same paper, Tan gave an example of a non-embeddable and non-Kählerian strongly pseudoconvex three-dimensional manifold withP ¹ as an exceptional set.     

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.     

Applications of Bruhat decompositions to complex hyperbolic geometry

The double coset space AΛ (n, ℂ) / U (n − 1, 1) is studied, where A consists of the diagonal matrices in GL (n, ℂ). This space naturally arises in the harmonic analysis on the hermitian symmetric space GL (n, ℂ) / U (n − 1, 1). It is shown here that these double cosets also represent a class of basic invariants related to complex hyperbolic geometry. An algebraic parametrization for the double cosets is given and it is shown how this may be used to conveniently compute the geometric invariants.     

Embedding open Riemann surfaces in Riemannian manifolds

For the Riemann surface of the topological type, we can get a conformai model in orientable Riemannian manifolds. We will prove that there is a conformally equivalent model in orientable Riemannian manifolds for a given open Riemann surface. To end up we utilize Garsia 's Continuity lemma and Brouwer's Fixed Point lemma along with the Teichmüller theory.     

The heat flow and harmonic maps between complete manifolds

We consider the existence, uniqueness and convergence for the long time solution to the harmonic map heat equation between two complete noncompact Riemannian manifolds, where the target manifold is assumed to have nonpositive curvature. As an application, we solve the Dirichlet problem at infinity for proper harmonic maps between two hyperbolic manifolds for a class of boundary maps. The boundary map under consideration has finite many points at which either it is not differentiable or has vanishing energy density.     

Neighborhoods of Leviq-convex domains

We consider bounded and q-complete domains in C ⁿ with C ² boundary. We show that they enjoy special q-convexity properties, i.e., they admit q-convex bounded exhaustion function and their closure do have a fundamental system of (q + 1)-complete neighborhoods. Moreover, we produce an example of a q- complete domain (which we called the q-worm) with smooth boundary whose closure does not possess a fundamental system of q-complete neighborhoods. Finally, extensions of these results to p-complete manifolds are given.     

On proper holomorphic self-maps of generalized complex ellipsoids

The purpose of this paper is to prove that every proper holomorphic self-mapping of a Reinhardt domain Ω in C ⁿ which is a generalization of a complex ellipsoid is biholomorphic. The main novelty of our result is that Ω is a domain in C ⁿ such that it is allowed to have a boundary point at which the Levi determinant has infinite order of vanishing.     

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.     

Sharp lipschitz estimates for operator $$\bar \partial _M $$ on a q-pseudoconcave CR manifoldon a q-pseudoconcave CR manifold

We construct integral operators R_r and H_r on a regular q-pseudoconcave CR manifoldM such that