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Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain III: open Riemann surfaces

Qi'an Guan, Zheng Yuan

TL;DR

The paper investigates when the concavity of minimal $L^2$ integrals with Lebesgue-measurable gain on open Riemann surfaces degenerates to linearity and connects this to equality in optimal jets $L^2$ extension, yielding a weighted Suita-type conjecture for analytic subsets. Building a framework that uses Green functions, universal covers, and multiplicative characters, it derives a precise linearity criterion for finite point sets and establishes sharp equality criteria for optimal jet extensions independent of the full Suita solution. It then provides a comprehensive one-dimensional characterization (Theorem \text{c:L2-1d-char}) and concrete examples, illustrating how geometric data encode extension optimality. The results advance the understanding of when equality can occur in jet extension problems and link these cases to explicit analytic and geometric data, with broad implications for Suita-type inequalities on open Riemann surfaces.

Abstract

In this article, we present a characterization of the concavity property of minimal $L^2$ integrals degenerating to linearity in the case of finite points on open Riemann surfaces. As an application, we give a characterization of the holding of equality in optimal jets $L^2$ extension problem from analytic subsets to open Riemann surfaces, which is a weighted jets version of Suita conjecture for analytic subsets.

Concavity property of minimal $L^2$ integrals with Lebesgue measurable gain III: open Riemann surfaces

TL;DR

The paper investigates when the concavity of minimal integrals with Lebesgue-measurable gain on open Riemann surfaces degenerates to linearity and connects this to equality in optimal jets extension, yielding a weighted Suita-type conjecture for analytic subsets. Building a framework that uses Green functions, universal covers, and multiplicative characters, it derives a precise linearity criterion for finite point sets and establishes sharp equality criteria for optimal jet extensions independent of the full Suita solution. It then provides a comprehensive one-dimensional characterization (Theorem \text{c:L2-1d-char}) and concrete examples, illustrating how geometric data encode extension optimality. The results advance the understanding of when equality can occur in jet extension problems and link these cases to explicit analytic and geometric data, with broad implications for Suita-type inequalities on open Riemann surfaces.

Abstract

In this article, we present a characterization of the concavity property of minimal integrals degenerating to linearity in the case of finite points on open Riemann surfaces. As an application, we give a characterization of the holding of equality in optimal jets extension problem from analytic subsets to open Riemann surfaces, which is a weighted jets version of Suita conjecture for analytic subsets.
Paper Structure (13 sections, 25 theorems, 130 equations)

This paper contains 13 sections, 25 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Concavity property of minimal $L^2$ integrals and optimal $L^2$ extension
  3. A characterization for linearity to hold on open Riemann surfaces
  4. Optimal jets $L^2$ extension problem and generalized Suita conjecture
  5. Some results on open Riemann surfaces
  6. Optimal jets $L^2$ extension on open Riemann surfaces
  7. The concavity property of minimal $L^2$ integrals
  8. The concavity property on weakly pseudoconvex Kähler manifolds
  9. Necessary conditions for linearity to hold on open Riemann surfaces
  10. Application: $L^2$ extension from infinite points to $\Omega$
  11. Proofs of Theorem \ref{['thm:m-points']} and Remark \ref{['rem:1.1']}
  12. Proofs of Theorem \ref{['c:L2-1d-char']} and Remark \ref{['rem:1.2']}
  13. Appendix: An example of Theorem \ref{['c:L2-1d-char']}

Key Result

Theorem 1.1

$G(-\log r)$ is a concave function on $(0,1)$.

Theorems & Definitions (47)

  • Theorem 1.1: G16
  • Definition 1.2
  • Theorem 1.3: GMY-concavity2, see also GMY-boundary2
  • Theorem 1.5: OT87
  • Conjecture 1.6: Suita Conjecture suita72
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • Theorem 1.11
  • Remark 1.12
  • ...and 37 more