Peter Ebenfelt - Böcker

This book presents many of the main developments of the past two decades in the study of real submanifolds in complex space, providing crucial background material for researchers and advanced graduate students. The techniques in this area borrow from real and complex analysis and partial differential equations, as well as from differential, algebraic, and analytical geometry. In turn, these latter areas have been enriched over the years by the study of problems in several complex variables addressed here. The authors, M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, include extensive preliminary material to make the book accessible to nonspecialists. One of the most important topics that the authors address here is the holomorphic extension of functions and mappings that satisfy the tangential Cauchy-Riemann equations on real submanifolds. They present the main results in this area with a novel and self-contained approach. The book also devotes considerable attention to the study of holomorphic mappings between real submanifolds, and proves finite determination of such mappings by their jets under some optimal assumptions.The authors also give a thorough comparison of the various nondegeneracy conditions for manifolds and mappings and present new geometric interpretations of these conditions. Throughout the book, Cauchy-Riemann vector fields and their orbits play a central role and are presented in a setting that is both general and elementary.

Several Complex Variables is a beautiful example of a ?eld requiring a wide rangeoftechniquescoming fromdiverseareasin Mathematics.Inthe lastdecades, many major breakthroughs depended in particular on methods coming from P- tial Di?erential Equations and Di?erential and Algebraic Geometry. In turn, S- eralComplexVariablesprovidedresultsandinsightswhichhavebeenoffundam- tal importance to these ?elds. This is in particular exempli?ed by the subject of Cauchy-Riemanngeometry,whichconcernsitselfbothwiththetangentialCauchy- Riemannequationsandtheuniquemixtureofrealandcomplexgeometrythatreal objects in a complex space enjoy. CR geometry blends techniques from algebraic geometry, contact geometry, complex analysis and PDEs; as a unique meeting point for some of these subjects, it shows evidence of the possible synergies of a fusion of the techniques from these ?elds. The interplay between PDE and Complex Analysis has its roots in Hans Lewy's famous example of a locally non solvable PDE. More recent work on PDE has been similarly inspired by examples from CR geometry.The application of analytic techniques in algebraic geometry has a long history; especially in recent - years, the analysis of the ?-operator has been a crucial tool in this ?eld. The - ?-operator remains one of the most important examples of a partial di?erential operator for which regularity of solutions under boundary constraints have been extensively studied. In that respect, CR geometry as well as algebraic geometry have helped to understand the subtle aspects of the problem, which is still at the heart of current research.