In this paper we are interested in showing an alternative and simple proof for small energy implies regularity to the Cauchy problem of radially symmetric wave maps from the \((1+2)\) -dimensional Minkowski space to an arbitrary smooth Riemannian manifold \(\mathcal {M}\subset \textbf{R}^n\) with bounded first derivatives of the second fundamental form. Then, combining the classical works of Struwe (Math Z 242:407–414, 2002; Calc Var Partial Differ Equ 16:431–437, 2003), which gave a simple proof for non-concentration of energy to the corresponding problem, a new proof which we think should it be simple can be provided to the result of global regularity for radially symmetric \((1+2)\) -dimensional wave maps, which was first obtained by Christodoulou and Tahvildar-Zadeh (Commun Pure Appl Math 46:1041–1091, 1993). Our method relies on basic energy estimates, based on a new div-curl type lemma developed by Zhou ((1 + 2)-dimensional radially symmetric wave maps revisit) and Wang-Zhou (Global well-posedness for radial extremal hypersurface equation in (1 + 3)-dimensional minkowski space-time in critical sobolev space; On proof of the Wei-Yue Ding’s conjecture for Schrödinger map flow).
