Biharmonic surfaces with flat normal bundle in the four-dimensional sphere

We prove that a proper biharmonic surface with flat normal bundle in the four-dimensional unit sphere S⁴ lies minimally in a small hypersphere of radius 1/√2. Since the surfaces with constant mean curvature and those with parallel normalized mean curvature vector field are already understood, the core of the paper is the non-existence of biharmonic surfaces with flat normal bundle in S⁴ whose mean curvature is not constant and whose normalized mean curvature vector field is not parallel. We first show, adapting an argument of Fu, Hong and Yang, that the gradient of the mean curvature is a principal direction. Then the problem reduces to a first order system of ordinary differential equations with a constraint, and we show that this system has no admissible solutions by an elimination argument.