Walker 3-Manifolds as Ricci Solitons: Harmonic and Biharmonic Potentials

This paper investigates Ricci solitons on 3-dimensional Walker manifolds equipped with the canonical metric $ g = 2\,dx\otimes dz + \varepsilon\,dy\otimes dy + f(x,y,z)\,dz\otimes dz, \quad \varepsilon = \pm 1, $ under the assumption that the potential function $h$ is either harmonic or biharmonic with respect to the Laplace--Beltrami operator $\Delta_g$. In the harmonic case, we prove that the gradient Ricci soliton equation forces the structure function $f$ to be at most quadratic in $x$, with the soliton constant $\lambda$ controlling the leading coefficient. On flat Walker manifolds, where the scalar curvature vanishes identically, we establish that every Ricci soliton of the form $(M, g, \nabla h + Y, \lambda)$, with $h$ harmonic and $\operatorname{div}(Y) = 0$, is necessarily steady, and we construct explicit infinite families of harmonic potentials realizing this conclusion. We further produce a broad class of harmonic functions through separation of variables, exponential and trigonometric ansatze, and a diagonal reduction. In the biharmonic case, we show that the gradient Ricci soliton condition is equivalent to the harmonicity of the scalar curvature function $\partial_{xx}f$ under $\Delta_g$, and to the existence of a smooth function $\varpi = \varpi(y,z)$ encoding an integration constraint on $f$. This three-way equivalence gives a precise geometric characterization of Ricci solitons with biharmonic potential on Walker manifolds. Throughout, explicit families of structure functions compatible with each soliton type are identified, and the steady, shrinking, and expanding regimes are distinguished in terms of the geometry of $f$.