We study holomorphic immersions of open Riemann surfaces into ${\mathbb C}^n$ whose derivative lies in a conical algebraic subvariety $A$ of ${\mathbb C}^n$ that is smooth away from the origin. Classical examples of such $A$-immersions include null curves in ${\mathbb C}^3$ which are closely related to minimal surfaces in ${\mathbb R}^3$, and null curves in $SL_2(\mathbb{C})$ that are related to Bryant surfaces. We establish a basic structure theorem for the set of all $A$-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that $A$ is irreducible and is not contained in any hyperplane, we show that every $A$-immersion can be approximated by $A$-embeddings; this holds in particular for null curves in ${\mathbb C}^3$. If in addition $A \setminus \{0\}$ is an Oka manifold, then $A$-immersions are shown to satisfy the Oka principle, including the Runge and the Mergelyan approximation theorems. Another version of the Oka principle holds when $A$ admits a smooth Oka hyperplane section. This lets us prove in particular that every open Riemann surface is biholomorphic to a properly embedded null curve in ${\mathbb C}^3$.
COBISS.SI-ID: 16655705
From Preface: This book is an attempt to present a coherent account of Oka theory, from the classical Oka-Grauert theory originating in the works of Kiyoshi Oka and Hans Grauert to the contemporary developments initiated by Mikhael Gromov. Chapter 1 contains some preparatory material, and Chapter 2 is a brief survey of Stein space theory. In Chapter 3 we construct open Stein neighborhoods of certain types of sets in complex spaces that are used in Oka theory. Chapter 4 contains an exposition of the theory of holomorphic automorphisms of Euclidean spaces and of the density property, a subject closely intertwined with our main theme. In Chapter 5 we develop Oka theory for stratified fiber bundles with Oka fibers (this includes the classical Oka-Grauert theory), and in Chapter 6 we treat Oka-Gromov theory for stratified subelliptic submersions over Stein spaces. Chapters 7 and 8 contain applications ranging from classical to the recent ones. In Chapter 8 we present results on regular holomorphic maps of Stein manifolds; highlights include the optimal embedding theorems for Stein manifolds and Stein spaces, proper holomorphic embeddings of some bordered Riemann surfaces into ▫${\mathbb C}^2$▫, and the construction of noncritical holomorphic functions, submersions and foliations on Stein manifolds. In Chapter 9 we explore implications of Seiberg-Witten theory to the geometry of Stein surfaces, and we present the Eliashberg-Gompf construction of Stein structures on manifolds with suitable handlebody decomposition.
COBISS.SI-ID: 16008025
Let $B$ be the open unit ball in ${\Bbb C}^2$. This paper deals with the analog of Hartogs' separate analyticity theorem for CR functions on the sphere $bB$. We prove such a theorem for functions in $C^\infty (bB)$: If $a, b \in \overline B$, $a \ne b$ and if $f \in C^\infty (bB)$ extends holomorphically into $B$ along any complex line passing through either $a$ or $b$, then $f$ extends holomorphically through $B$. On the other hand, for each $k \in \Bbb{N}$ there is a function $f \in C^k(bB)$ which extends holomorphically into $B$ along any complex line passing through either $a$ or $b$ yet $f$ does not extend holomorphically through $B$. More generally, in the paper we obtain a fairly complete description of pairs of points $a, b \in {\Bbb C}^2$, $a \ne b$, such that if $f \in C^\infty (bB)$ extends holomorphically into $B$ along every complex line passing through either $a$ or $b$ that meets $B$, then $f$ extends holomorphically through $B$.
COBISS.SI-ID: 16521561
In this paper we give an extension of the Cartier-Gabriel-Kostant structure theorem to Hopf algebroids.
COBISS.SI-ID: 16432473
By using an explicit Bellman function, we prove a bilinear embedding theorem for the Laplacian associated with a weighted Riemannian manifold $(M,\mu_\varphi)$ having the Bakry-Emery curvature bounded from below. The embedding, acting on the cartesian product of $L^p(M,\mu_\varphi)$ and $L^q(T^ast M, \mu_\varphi)$, $1/p + 1/q = 1$, involves estimates which are independent of the dimension of the manifold and linear in $p$. As a consequence we obtain linear dimension-free estimates of the $L^p$ norms of the corresponding shifted Riesz transform. All our proofs are analytic.
COBISS.SI-ID: 16719705