A curvature estimate for holomophic maps on open Riemann surfaces

TL;DR

This paper uses jet differentials to derive a Gauss curvature estimate for open Riemann surfaces with metrics from holomorphic maps that are highly ramified over high-degree hypersurfaces.

Contribution

It introduces a novel application of jet differentials to estimate curvature on open Riemann surfaces in the context of highly ramified holomorphic maps.

Findings

Gauss curvature estimate established for specific holomorphic maps

Application of jet differentials technique to curvature problems

Results applicable to high-degree hypersurfaces

Abstract

We apply the technique of jet differentials to establish a Gauss curvature estimate for an open Riemann surface $M$, equipped with a conformal metric induced from a nonconstant holomorphic map that is highly ramified over a generic hypersurface of sufficiently high degree.