Uniqueness of the zero of u0 on compact harmonic manifolds

Determine whether, on any compact harmonic manifold (i.e., Ric > 0), the solution u0 of the ODE u0''(r) + h(r) u0'(r) + (n/(n−1)) Ric · u0(r) = 0 with initial conditions u0(0) = 1 and u0'(0) = 0 has at most one zero in the interval [0, diam(M)].

Background

The paper introduces the radial function u0 as the unique solution of a second-order linear ODE depending on the mean curvature h and Ricci curvature on a harmonic manifold. Regularity and sign properties of u0 are crucial in defining auxiliary functions (Fv) used to reduce the vector equation to a solvable form.

For non-compact (Ric ≤ 0) harmonic manifolds, the authors establish that u0 does not change sign. In the compact (Ric > 0) case, to proceed they assume u0 has a unique zero point strictly inside [0, diam(M)], but they acknowledge that, beyond the sphere, they lack a general proof of this property.