Laplacian Vanishing Theorem for Quantized Singular Liouville Equation

TL;DR

This paper proves a new vanishing theorem for singular Liouville equations with quantized sources, providing second order estimates crucial for understanding complex blow-up behaviors.

Contribution

It introduces the first second order estimates for Liouville equations with quantized sources and non-simple blow-ups, advancing theoretical understanding.

Findings

Laplacian of coefficient tends to zero in non-simple blow-ups

First second order estimates for quantized singular Liouville equations

Key ideas applicable to various problems in the field

Abstract

In this article we establish a vanishing theorem for singular Liouville equation with quantized singular source. If a blowup sequence tends to infinity near a quantized singular source and the blowup solutions violate the spherical Harnack inequality around the singular source (non-simple blow-ups), the Laplacian of a coefficient function must tend to zero. This seems to be the first second order estimates for Liouville equation with quantized sources and non-simple blow-ups. This result as well as the key ideas of the proof would be extremely useful for various applications.