Cauchy-Pompeiu type formulas for d-bar on affine algebraic Riemann surfaces and some applications

TL;DR

This paper develops Cauchy-Pompeiu formulas for the d-bar operator on affine algebraic Riemann surfaces, constructs Faddeev-Green functions, and extends conductivity reconstruction methods from complex to algebraic Riemann surfaces.

Contribution

It introduces explicit formulas for d-bar on affine algebraic Riemann surfaces and extends conductivity reconstruction techniques to these surfaces.

Findings

Explicit formulas for d-bar on affine algebraic Riemann surfaces.

Construction of Faddeev-Green functions for Laplacian on these surfaces.

Extension of conductivity reconstruction methods to algebraic Riemann surfaces.

Abstract

We have obtained the explicit versions and precisions for the Hodge-Riemann decomposition of formes on affine algebraic curve V. The main application consists in the construction of Faddeev-Green function for Laplacian on V. Basing on this [HM](arXiv:0804.3951 and J.Geom.Anal., 2008,18), we extended from the case X in C to the case of bordered Riemann surface X in V the R.Novikov (1988) scheme for the effective reconstruction of conductivity function sigma on X through Dirichlet-to-Neumann mapping on bX for solutions of d(sigma d^cU)=0. In Sec.4 we give a correction of the paper [HM].