Hyperbolic manifolds whose envelopes of holomorphy are not hyperbolic

Schlicht envelopes of holomorphy and foliations by lines

Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope…

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a nei…

Envelopes of holomorphy and extension of functions of bounded type

We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not…

A new Hartogs type extension results for the cross-like objects

We discuss the problem of the existence of envelopes of holomorphy of the A-crosses, which leads us to the far-reaching generalizations of the famous Hartogs theorem. We also take under consideration the issue of the existence of "nice" general descriptions of envelopes of holomorphy of the cross-like objects in terms of the relative extremal function, which seems to be very natural in the light of the extension theory of separately holomorphic functions on classical crosses and (N,k)-crosses.

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlevé problem; metrically thin …

复 Grassmann 流形中 Matsuki 轨道的全纯包络

Envelope of Holomorphy of Matsuki Orbits in Complex Grassmannians

本文利用 Rossi 全纯延拓理论与全纯纤维丛结构，证明复 Grassmann 流形中 Matsuki 轨道的全纯包络双全纯等价于包含它的 $K$-轨道。

On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat ``hat''. In our main theorem, we show that every C^infty-smooth CR diffeomorphism h: M to M' between two globally minimal real analytic hypersurfaces in C^n (n >1) is real analytic at every point of M if M' is holomorphically nondegenerate. More generally, we establish th…

Plurisubharmonicity of envelopes of disc functionals on manifolds

We show that a disc functional on a complex manifold has a plurisubharmonic envelope if all its pullbacks by holomorphic submersions from domains of holomorphy in affine space do and it is locally bounded above and upper semicontinuous in a certain weak sense. For naturally defined classes of disc functionals on manifolds, this result reduces a property somewhat stronger than having a plurisubharmonic envelope to the affine case. The proof uses a recent Stein neighbourhood construction of Rosay, who proved the plurisubharmonicity of the Poisson envelope on all manifolds. As a consequence, the …

具有多叶包络的截断管状域

Truncated tube domains with multi-sheeted envelope

本文通过构造同胚于4维球的反例，否定了截断管状域的全纯包络总是单叶的猜想，并证明包络可具有任意多叶，同时给出复二维下单叶性的充分条件。

Characteristic foliations on maximally real submanifolds of C^n and envelopes of holomorphy

Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at finitely many complex tangencies which are hyperbolic in the sense of E. Bishop and if the union of separatrices is a tree of curves without cycles, we show that every compact K of S is CR-, W- and L^p-removable (Theorem~1.3). We treat this seemingly global problem by means of purely local techniques, namely by means of families of small analytic discs partially …

高维复空间中截断管状域与特殊域的包络

Envelope of truncated tubes and special domains in higher complex dimensions

本文引入特殊域并证明拟凸截断管状域均为特殊域，进而将 Jarnicki-Pflug 关于截断管状域全纯包络的结果推广到 $\C^n$ 中特殊域的包络，同时给出复二维单叶性结果及其两个高维推广。

Some results related to group actions in several complex variables

In this talk, we'll present some recent results related to group actions in several complex variables. We'll not aim at giving a complete survey about the topic but giving some our own results and related ones. We'll divide the results into two cases: compact and noncompact transformation groups. We emphasize some essential differences between the two cases. In the compact case, we'll mention some results about schlichtness of envelopes of holomorphy and compactness of automorphism groups of some invariant domains. In the noncompact case, we'll present our solution of the longstanding problem …

Topological criteria for schlichtness

We give two sufficient criteria for schlichtness of envelopes of holomorphy in terms of topology. These are weakened converses of results of Kerner and Royden. Our first criterion generalizes a result of Hammond in dimension 2. Along the way we also prove a generalization of Royden's theorem.

Effective actions of the general indefinite unitary groups on holomorphically separable manifolds

We determine all holomorphically separable complex manifolds of dimension $p+q$ which admits a smooth envelope of holomorphy such that the general indefinite unitary group of size $p+q$ acts effectively by holomorphic transformations. Also we give exact description of the automorphism groups of those complex manifolds. As an application we consider a characterization of those complex manifolds by their automorphism groups.

Extension Properties of Meromorphic Mappings with Values in Non-Kahler Manifolds

We prove an analogue of E. Levi's Continuity Principle for meromorphic mappings with values in arbitrary compact complex manifolds in place of the Riemann sphere $\cc\pp^1$. The result is achieved by introducing a new extension method for meromorphic mappings. One of the corollaries reads as follows: If a compact complex surface $X$ is not "among the known ones" then for every domain $Ω$ in a Stein surface every meromorphic mapping $f:Ω\to X$ is in fact holomorphic and extends as a holomorphic mapping $\hat f:\hat D\to X$ of the envelope of holomorphy $\hat D$ of $D$ into $X$. In this last ver…

Applications of Envelopes

Intuitively, an envelope of a family of curves is a curve that is tangent to a member of the family at each point. Here we use envelopes of families of circles to study objects from matrix theory and hyperbolic geometry. First we explore relationships between numerical ranges of $2\times 2$ matrices and families of circles to study the elliptical range theorem. Then we deduce a relationship between envelopes and the boundaries of families of intersecting circles and use it to find the boundaries of various families of pseudohyperbolic disks.

双曲3-空间中极限球的包络与Weingarten曲面

Envelopes of Horospheres and Weingarten Surfaces in Hyperbolic 3-Space

本文推导双曲空间中极限球包络曲面的微分几何公式，证明H^3中Weingarten曲面的存在性与正则性，并揭示其与共形映射的深刻联系。