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 $Q_g$ 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

References

[1] Axler, S., Bourdon, P., Ramey, W.: Harmonic function theory. Second edition. Graduate Texts in Mathematics, 137. Springer-Verlag, New York, (2001)

[2] Branson T, Gilkey P, Pohjanpelto J. Invariants of locally conformally flat manifolds. Trans. Amer. Math. Soc.. 1995, 347 3 939-953

[3] Buzano R, Nguyen H. The higher-dimensional Chern-Gauss-Bonnet formula for singular conformally flat manifolds. J. Geom. Anal.. 2019, 29 2 1043-1074

[4] Buzano R, Nguyen H. The The Chern-Gauss-Bonnet formula for singular non-compact four-dimensional manifolds. Comm. Anal. Geom.. 2019, 27 8 1697-1736

[5] Chang S-YA, Qing J, Yang PC. On the Chern-Gauss-Bonnet integral for conformal metrics on $R^4$. Duke Math. J.. 2000, 103 3 523-544

[6] Fang H. On a conformal Gauss-Bonnet-Chern inequality for LCF manifolds and related topics. Calc. Var. Partial Diff. Equa.. 2005, 23 4 469-496

[7] Huber A. On subharmonic functions and differential geometry in the large. Comment. Math. Helv.. 1957, 32 13-72

[8] Kuwert E, Li YX. $W^{2,2}$-conformal immersions of a closed Riemann surface into${\mathbb{R} }^n$. Comm. Anal. Geom.. 2012, 20 2 313-340

[9] Li, Y.X., Chen, B.: Huber’s theorem for manifolds with $L^{\frac{n}{2}}$ integrable Ricci curvatures. arXiv:2111.07120

[10] Li, Y.X., Wang, Z. H.: Manifolds for which Huber’s Theorem holds. arXiv:2108.06708

[11] Li YX, Zhou ZP. Conformal metric sequences with integral-bounded scalar curvature. Math. Z.. 2020, 295 3–4 1443-1473

[12] Müller S, Šverák V. On surfaces of finite total curvature. J. Differ. Geom.. 1995, 42 2 229-258

[13] Simon L. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems. Ann. Math.. 1983, 118 3 525-571

[14] Wang SW, Wang Y. Integrability of scalar curvature and normal metric on conformally flat manifolds. J. Diff. Equa.. 2018, 265 4 1353-1370

[15] Wang Y. The isoperimetric inequality and quasiconformal maps on manifolds with finite total Q-curvature. Int. Math. Res. Not.. 2012, 2 394-422

[16] Xu, K.: Compactness of isospeectral conformal metrics on $4$-manifolds. arXiv:1911.13100

Funding

Beijing Municipal Education Commission(KM202310028014)