Complex Analysis: Several Complex Variables and Connections with PDE Theory and Geometry

ISBN-13: 9783034600088 / Angielski / Twarda / 2010 / 340 str.

Ebenfelt

Opis

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."

Spis treści

Preface.- A mathematical CV of Linda Rothschild: Her contributions to complex analysis.- Oblique polar lines of RX |f|2|g|2µ.- On involutive systems of first-order nonlinear PDEs.- Gevrey Hypoellipticity for an interesting variant of Kohn’s operator.- Subelliptic Estimates.- Invariant CR Mappings.- On the subellipticity of some hypoelliptic quasihomogeneous systems of complex vector fields.- Invariance of the parametric Oka property.- Positivity of the #x2202;-Neumann Laplacian.- Compactness estimates for the #x2202;-Neumann problem in weighted L2-spaces.- Remarks on the homogeneous complex Monge-Ampère equation.- A Radó theorem for locally solvable structures of co-rank one.- Applications of a parametric Oka principle for liftings.- Stability of the vanishing of the #x2202;b-cohomology under small horizontal perturbations of the CR structure in compact abstract q-concave CR manifolds.- coherent Sheaves and Cohesive Sheaves.- Characteristic classes of the boundary of a complex b-manifold.- Solvability of planar complex vector fields with applications to deformation of surfaces.- The Gauss map on complex hyperbolic space forms.- The large time asymptotics of the entropy.- The closed range property for #x2202; on domains with pseudoconcave boundary.- New normal forms for Levi-nondegenerate hypersurfaces.

Blurb

This is a reference book for academics working in the fields of several complex variables, PDE theory and geometry. The book collects articles on most recent developments in these fields with special emphasis on interactions between them. It includes introductory and expository presentations of the relevant tools, techniques and objects for graduate and PhD students with a corresponding specialization. The volume comprises contributions of leading experts in the field written in a comprehensive and accessible style. Contributors: D. Barlet, S. Berhanu, A. Bove, D.W. Catlin, J.P. D’Angelo, M. Derridj, F. Forstnerič, S. Fu, K. Gansberger, P. Guan, F. Haslinger, B. Helffer, J. Hounie, N. Hungerbühler, F. Lárusson, Ch. Laurent-Thiébaut, L. Lempert, H.-M. Maire, G.A. Mendoza, A. Meziani, N. Mok, M. Mughetti L. Ni, M.-C. Shaw, D.S. Tartakoff, D. Zaitsev