Abstract:

The famous Cheng-Yau gradient estimate implies that on a complete Riemannian manifold with nonnegative Ricci curvature, any harmonic function that grows sublinearly must be a constant. This is the same as saying the function is closed as a 0-form. We prove an analogous result for harmonic 1-forms. Namely, on a complete Ricci-flat manifold with Euclidean volume growth, any harmonic 1-form with polynomial sublinear growth must be the differential of a harmonic function. We prove this by proving an L^2 version of the "gradient estimate" for harmoinc 1-forms. As a corollary, we show that when the manifold is Ricci-flat Kähler with Euclidean volume growth, then any subquadratic harmonic function function must be pluriharmonic. This generalizes a result of Conlon-Hein.