The gradient of certain harmonic functions on manifolds of almost nonnegative Ricci curvature (Q1601477)

Let \(\gamma(t)\), \(0\leq t\leq L\) be a minimal geodesic in the Riemannian manifold \(M^n\) with almost non-negative Ricci curvature \(Ric \geq - (n-1)H\), and \(p\) a point such that \(\text{dist}(p,\gamma(t))< \delta L\) for some \(L/3 < t < 2L/3\). In the ball \(B_L(p)\) define a harmonic function \(b\) whose boundary values equal \(\text{dist}(x,\gamma(0))\) for \(x\in \partial B_L(p)\). In the paper under review the author proves that the gradient \(\nabla b\) of the function \(b\) does not vanish if \(L^{-1}, H, \delta\) are sufficiently small and the ball \(B_L(p)\) is close enough in the Gromov-Hausdorff distance to the Euclidean ball of the same radius.