We prove the parametric homotopy principle for holomorphic immersions of Stein manifolds into Euclidian space and the homotopy principle with approximation on holomorphically convex sets. We write an integration by parts like formula for the solution ▫$f$▫ to the problem ▫$Lf|_\Sigma=g$▫, where ▫$L$▫ is a holomorphic vector field, semi-transversal to analytic variety ▫$\Sigma$▫.
COBISS.SI-ID: 15890777
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
We provide a proof of sharp lower ▫$L^p$▫ bounds for powers of the Ahlfors-Beurling operator ▫$T$▫ and improve the numerical constant in their asymptotic estimates. We also discuss the possibilities of applying de Branges' theorem in the ▫$p-1$▫ problem.
COBISS.SI-ID: 16090713