In this paper, we introduce and prove three analytic results related to uniform convergence, properties of Newtonian potential, and convergence of sequences in Sobolev space constrained by their Laplacian. Then, utilizing our analytic results, we develop a complete proof of a crucial estimate appearing in the results of Guofang Wang and Xiaohua Zhu, which states the classification of extremal Hermitian metrics with finite energy and area on compact Riemann surfaces and finite singularities satisfying small singular angles.
SANG Haoxin
,
WU Yingyi
. Some analytic results and applications in extremal Hermitian metrics[J]. Journal of University of Chinese Academy of Sciences, 2025
, 42(5)
: 589
-599
.
DOI: 10.7523/j.ucas.2023.091
[1] Chen X X. Obstruction to the existence of metric whose curvature has umbilical hessian in a K-surface[J]. Communications in Analysis and Geometry, 2000, 8(2): 267-299. DOI:10.4310/CAG.2000.v8.n2.a2.
[2] Chen X X. Extremal Hermitian metrics on Riemann surfaces[J]. Calculus of Variations and Partial Differential Equations, 1999, 8(3): 191-232. DOI:10.1007/s005260050123.
[3] Wang G F, Zhu X H. Extremal Hermitian metrics on Riemann surfaces with singularities[J]. Duke Mathematical Journal, 2000, 104(2). DOI:10.1215/S0012-7094-00-10421-8.
[4] Chen X X. Weak limits of Riemannian metrics in surfaces with integral curvature bound[J]. Calculus of Variations and Partial Differential Equations, 1998, 6(3): 189-226. DOI:10.1007/s005260050089.
[5] Wu Y Y. On the character 1-form of an HCMU metric[J]. Journal of the Graduate School of the Chinese Academy of Sciences, 2009, 26(2): 185-193. DOI:10.7523/j.issn.2095-6134.2009.2.006.
[6] Adams R A, Fournier J J F. Sobolev spaces[M]. 2nd ed. Amsterdam: Academic Press, 2003.
[7] Evans L C. Partial differential equations[M]. Providence, R.I.: American Mathematical Society, 1998.
[8] Donaldson S K. Riemann surfaces[M]. Oxford: Oxford University Press, 2011.
[9] Gilbarg D, Trudinger N S. Elliptic partial differential equations of second order[M]. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. DOI:10.1007/978-3-642-61798-0.
[10] Chen Q, Wu Y Y. Character 1-form and the existence of an HCMU metric[J]. Mathematische Annalen, 2011, 351(2): 327-345. DOI:10.1007/s00208-010-0598-z.
[11] Chen Q, Wu Y Y, Xu B. On one-dimensional and singular Calabi’s extremal metrics whose Gauss curvatures have nonzero umbilical Hessians[J]. Israel Journal of Mathematics, 2015, 208(1): 385-412. DOI:10.1007/s11856-015-1204-6.
