Pole Regularity of Green function and rigidity

Abstract

Let G(p, .) be the normalized minimal positive Green function on a complete nonparabolic Riemannian manifold with nonnegative Ricci curvature, and set b = G^(1/(2-n)). Colding established the following rigidity result in dimension three: if |nabla b|^2 admits a C^2 regularity across the pole, then the manifold must be isometric to Euclidean space. We show that rigidity conclusions corresponding to the pole regularity condition alone do not hold in higher dimensions: for any n >= 4, we can construct explicit complete nonflat rotationally symmetric manifolds with nonnegative sectional curvature and Euclidean volume growth, such that |nabla b|^2 extends smoothly across the pole. For any odd integer n >= 3, we impose finite-order pointwise curvature conditions at the pole (where this condition is vacuous when n = 3), so that the Green function satisfies the asymptotic expansion G(p,x) = r^(2-n) + H_p + O_1(r). Under this curvature assumption, if |nabla b|^2 extends to a C^(n-2) function near the pole, then the manifold must be isometric to R^n, and this regularity threshold is also sharp. As special cases: in dimension three, if |nabla b|^2 is in C^1, then the manifold must be isometric to Euclidean space; in dimension five, it suffices to assume that the scalar curvature at the pole satisfies Scal(p) = 0 and |nabla b|^2 is in C^3. In contrast, for any even dimension n >= 4, there exist complete nonflat examples that are Euclidean in a neighborhood of the pole and admit a smooth extension of |nabla b|^2 across the pole.