MathCCSセミナー 2025/11/10

日時 / Date

2025年11月10日(月) 14:30～15:30 / November 10th (Mon) 2025 14:30-15:30

場所 / Venue

レジリエント社会構築イノベーションセンター 3階大会議室 /
Main Conference Room, 3rd Floor, Innovation Center for Creation of a Resilient Society

言語 / Language

英語/ English

Speaker

宮武夏雄（東北大学 MathCCS）/ Natsuo Miyatake (Tohoku University MathCCS)

Title

Complete harmonic metrics and subharmonic functions

Abstract

Let $X$ be a Riemann surface, $K_X \to X$ its canonical bundle, and $r \geq 2$ a natural number. One can then define the notion of a cyclic Higgs bundle associated with a holomorphic section $q \in H^0(K_X^r)$ and a square root of $K_X \to X$. Furthermore, harmonic metrics on the cyclic Higgs bundle can be defined as solutions to an elliptic equation constructed from the holomorphic section $q$. In 2020, Li-Mochizuki introduced the concept of completeness as an asymptotic condition for harmonic metrics on cyclic Higgs bundles, and established both the existence and uniqueness of the complete harmonic metrics for any non-zero section $q$. Beyond the original motivations and background of Li-Mochizuki in introducing complete harmonic metrics, I expect that they may lead to further developments through their connections with complex analysis and potential theory. In this talk, I will explain these ideas based on the following three preprints:

• N. Miyatake, Shannon entropy for harmonic metrics on cyclic Higgs bundles, arXiv: 2410.08571.
• N. Miyatake, Shannon entropy for harmonic metrics on cyclic Higgs bundles II, arXiv: 2508.12844.
• N. Miyatake, Complete harmonic metrics and subharmonic functions on the unit disc, arXiv: 2508.12848.