Rotationally symmetric symphonic maps

We consider a functional of pullbacks of metrics on the space of maps f between Riemannian manifolds. Harmonic maps are stationary points of the energy functional E ( f ) which is an integral of the trace of the pullback of the metric of the target manifold by f . Our functional E_sym (f) is an integral of the norm of the pullback. Stationary maps for E_sym(f) are called as symphonic maps (Kawai in Nonlinear Anal. 74: 2284-2295, 2011), (Kawai in Differ. Geom. Appl. 44: 161-177, 2016), (Misawa in Nonlinear Anal. 75: 5971-5974, 2012), (Misawa in Calc. Var. Part. Differ. Equ. 55: 1-20, 2016), (Misawa in Adv. Differ. Equ. 23: 693-724, 2018), (Misawa in Equ. Appl. 2: 1-20, 2021), (Misawa in Adv. Geom. 22: 23-31, 2022), (Nakauchi in Nonlinear Anal. 108: 87-98, 2014) and (Nakauchi in Ricerche di Matematica 60: 219-235, 2011). In this paper, we are concerned with rotationally symmetric maps. We prove that any rotationally symmetric map between 4-dimensional model spaces is a symphonic map if and only if it is a conformal map.