Connectedness properties of the space of complete nonnegatively curved planes

Abstract

We study the space of complete Riemannian metrics of nonnegative curvature on the plane equipped with the \(C^k\) topology. If \(k\) is infinite, we show that the space is homeomorphic to the separable Hilbert space. For any \(k\) we prove that the space cannot be made disconnected by removing a finite dimensional subset. A similar result holds for the associated moduli space. The proof combines properties of subharmonic functions with results of infinite dimensional topology and dimension theory. A key step is a characterization of the conformal factors that make the standard Euclidean metric on the plane into a complete metric of nonnegative sectional curvature.