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A curvature estimate for holomophic maps on open Riemann surfaces

Yunling Chen, Dinh Tuan Huynh

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.

A curvature estimate for holomophic maps on open Riemann surfaces

Abstract

We apply the technique of jet differentials to establish a Gauss curvature estimate for an open Riemann surface , equipped with a conformal metric induced from a nonconstant holomorphic map that is highly ramified over a generic hypersurface of sufficiently high degree.
Paper Structure (10 sections, 17 theorems, 93 equations)

This paper contains 10 sections, 17 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Second Main Theorem from logarithmic jet differentials
  3. Logarithmic jet differentials
  4. Second Main Theorem
  5. Existence of logarithmic jet differentials with high vanishing order
  6. Non-integrated defect relation from logarithmic jet differentials
  7. A curvature estimate
  8. Normal family of holomorphic maps
  9. Technical lemmas
  10. Proof of Theorem \ref{['gauss curvature estimate']}

Key Result

Theorem 1.1

Let $M$ be an open Riemann surface and let $f\colon M\rightarrow\mathbb{CP}^n$ be a non-constant holomorphic map. For a positive integer $p$, consider the conformal metric on $M$ given by where $F$ is a reduced representation of $f$ and $\omega$ is a holomorphic $1$--form on $M$. Let $D\subset\mathbb{CP}^n$ be a generic Kobayashi hyperbolic hypersurface of degree $d$ such that $f(M)\not\subset D$

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1: First Main Theorem
  • Theorem 2.2: Logarithmic Derivative Lemma Ru2023
  • Theorem 2.3: Second Main Theorem
  • proof
  • Corollary 2.4: Defect relation
  • proof
  • Corollary 2.5: Ramification estimate
  • proof
  • ...and 19 more