Abstract: 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 Guofang Wang and Xiaohua Zhu's results, which states the classification of extremal Hermitian metrics with finite energy and area on compact Riemann surfaces and finite singularities satisfying small singular angles.

Key words: extremal Hermitian metrics, conical singularities of metrics, Newtonian potential