Critical $\bar{\partial}$ problems in one complex dimension and some remarks on conformally invariant variational problems in two real dimensions

TL;DR

This paper investigates optimal conditions for solving a linear first-order system on vector bundles over Riemann surfaces, leading to new insights on regularity of harmonic maps and conformally invariant variational problems in two dimensions.

Contribution

It extends classical results on holomorphic structures to non-smooth connections, providing a short proof of harmonic map regularity and re-proving recent estimates in conformal variational problems.

Findings

Established optimal conditions for connection forms to ensure holomorphic frames.

Proved regularity of harmonic maps in two dimensions.

Re-proved a recent estimate on conformally invariant variational problems.

Abstract

We will study a linear first order system, a connection $\db$ problem, on a vector bundle equipped with a connection, over a Riemann surface. We show optimal conditions on the connection forms which allow one to find a holomorphic frame, or in other words to prove the optimal regularity of our solution. The underlying geometric principle, Theorem \ref{theorem KM}, is classical and well known \cite[Theorem 1]{KM}; it gives necessary and sufficient conditions for a connection to induce a holomorphic structure on a vector bundle over a complex manifold. Here we explore the limits of this statement when the connection is not smooth and our findings lead to a very short proof of the regularity of harmonic maps in two dimensions as well as re-proving a recent estimate of Lamm and Lin \cite{lamm_lin} concerning conformally invariant variational problems in two dimensions.