Application of metric currents to complex analysis

TL;DR

This paper extends the theory of metric currents to complex analysis and singular spaces, introducing new concepts like bidimension and Dolbeault decomposition, and applies this to solve Cauchy-Riemann equations in various complex settings.

Contribution

It develops a Sobolev and L2 theory for metric currents on singular spaces and adapts complex analysis tools to infinite-dimensional spaces, broadening the scope of the field.

Findings

Characterization of holomorphic currents in metric spaces

Solution of Cauchy-Riemann equations on singular and infinite-dimensional spaces

Development of a capacity theory for singular sets

Abstract

In 2000, Ambrosio and Kirchheim, with the paper "Currents in metric spaces", settled the foundations of a theory of currents on metric spaces and used it to pose and solve Plateau's problem in a wide class of Banach spaces. Following an idea of De Giorgi, they gave a new definition of current which was meaningful on any metric space. A metric current is a multilinear functional on (k+1)-tuples of Lipschitz functions (with the first one bounded) satisfying a continuity property, a locality property and a finite mass property. The second chapter of the thesis is devoted to the theory of local metric currents. We introduce the basic concepts and adapt the theory to the complex case, defining the bidimension, the Dolbeault decomposition and related notions. A characterization of holomorphic currents is given. The development of a Sobolev theory on singular space is our main concern in the third chapter. We give a characterization of Sobolev functions in terms of their behavior and growth on the regular part; this leads to a capacity theory for the singular set which allows us to obtain an approximation result, using functions with support in the regular part. The second part of the chapter deals with the L2 theory on singular spaces. In the fourth chapter, some applications of the theory are discussed. We solve Cauchy-Riemann equation on completely reducible singularities, by means of a structure theorem for metric currents; we also treat the equation in Lp on complex curves and outline a possible approach for the same problem on complex spaces which can be embedded as divisors in C^n. The final chapter tries to spread some light on the complex geometry in infinite dimensional spaces. After solving the Cauchy-Riemann equation in Banach spaces, in terms of the quasi-local metric currents, we turn to the study of positive currents, holomorphic chains and their boundaries in Hilbert spaces.