Three Circles Theorem for Volume of Conformal Metrics

Abstract

In this paper, we establish three circles theorem for volume of conformal metrics whose scalar curvatures are integrable in a critical (scaling invariant) norm. As applications, we analyze the asymptotic behavior of such metrics near isolated singularities and use it to show the residual terms of the Chern–Gauss–Bonnet formula are integers. Such strong rigidity implies a vanishing theorem on the integral value of the

Q_g

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q_g$$\end{document}

curvature, with application to the bi-Lipschitz equivalence problem for conformal metrics.

Keywords

Three Circles Theorem / Conformal metric / Critical integral of Curvature / Bi-Lipschitz regularity / 53C21 / 53C18 / 58J05

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Funding

Beijing Municipal Education Commission(KM202310028014)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature