Summary: The problem of finding \(p\)-harmonic flows arises in a wide range of applications including color image (chromaticity) denoising, micromagnetics, liquid crystal theory, and directional diffusion. In this paper, we propose an innovative curvilinear search method for minimizing \(p\)-harmonic energies over spheres. Starting from a flow (map) on the unit sphere, our method searches along a curve that lies on the sphere in a manner similar to that of a standard inexact line search descent method. We show that our method is globally convergent if the step length satisfies the Armijo-Wolfe conditions. Computational tests are presented to demonstrate the efficiency of the proposed method and a variant of it that uses Barzilai-Borwein steps.