3-circle Theorem for Willmore surface I

TL;DR

This paper uses the 3-circle theorem to analyze the blow-up behavior of Willmore surfaces, providing decay estimates and new proofs for key results like energy quantization, removable singularities, and gap theorems.

Contribution

It introduces a novel application of the 3-circle theorem to Willmore surfaces, offering streamlined proofs and decay estimates for the second fundamental form.

Findings

Decay estimate of the second fundamental form along neck regions

Simplified proofs of energy identity and quantization

Establishment of removable singularities and gap theorems

Abstract

In this paper, we study the blow-up of Willmore surfaces. By using the 3-circle theorem, we prove a decay estimate of the second fundamental form along the neck region. This estimate provides a new perspective and streamlined proofs to a few key results in this field, such as the energy identity(quantization), removable singularities and gap theorem.