Linear growth harmonic functions on complete manifolds with nonnegative Ricci curvature (Q1913613)

The main result of this paper is that for a complete Riemannian manifold \(M^n\) with nonnegative Ricci curvature any tangent cone at infinity, \(M_\infty\), splits isometrically as \(\underline{M}_\infty\times \mathbb{R}^k\), where \(\mathbb{R}^k\) has its standard flat metric, provided that the space of linear growth harmonic functions has dimension \(k+1\). A consequence of the theorem is that \(M^n\) is isometric to \(\mathbb{R}^n\) if the space of linear growth harmonic functions has dimension \(n+1\).