Approximation by Meromorphic k-Differentials on Compact Riemann Surfaces

Abstract

The main theorem of this article is a Runge type theorem proved for k-differentials \((k\ge 2)\). The integrability in the \(L^1\)- norm is defined for k-differentials in Section 2. We consider k-differentials which are integrable in the defined \(L^1\)- norm on the Riemann surface, and are holomorphic on an open subset of that surface. We will show those k-differentials can be approximated by meromorphic k-differentials. The proof applies a generalized form of the Poincaré series map. This generalized form is proved in Section 3. Section 2 contains the definition of the Poincaré series and its convergence, with particular focus on the convergence of the Poincaré series for rational functions, which is applied in the main theorem. Sections 3 and 4 contain the new results proved in this paper. The statement and proof of the main theorem are in Section 4.