Mathematics – Complex Variables
Scientific paper
2011-12-07
Mathematics
Complex Variables
Scientific paper
The aim of this paper is to show two applications of metric currents to complex analysis. After recalling the basic definitions, we give a detailed proof of the comparison theorem between metric currents and classical ones on a manifold. In Section 3 we introduce the concept of bidimension for a metric current on a finite dimensional space, showing that the usual properties of $(p,q)-$currents still hold, except for the existence of a Dolbeault decomposition. Section 4 is devoted to the analysis of a particular class of complex spaces, whose structure allows us to give a structure theorem for currents, solve the Cauchy-Riemann equation and characterize holomorphic currents. In Section 5, we introduce the concept of bidimension of (global) metric currents on a Banach space and relate it to the behaviour of the finite dimensional projections of the currents. In section 6 we define a new class of currents, the \emph{quasi-local} metric currents, which are usual metric currents when restricted to bounded sets, and we give a definition of $(p,q)-$current in this new class. The last Section shows how to employ these newly defined quasi-local currents in order to obtain a solution to the equation $\debar U=T$, when $T$ is of bidimension $(0,q)$ and its support is bounded; finally, we extend the result to a current $T$ with generic bidimension, $\debar-$closed, with bounded support.
No associations
LandOfFree
Some applications of metric currents to complex analysis does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Some applications of metric currents to complex analysis, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Some applications of metric currents to complex analysis will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-593990