Connectedness properties of the space of complete nonnegatively curved planes

TL;DR

This paper explores the topological structure of the space of complete nonnegatively curved metrics on the plane, revealing it is homeomorphic to a Hilbert space for infinite smoothness and has robust connectedness properties.

Contribution

It characterizes the topology of the space of such metrics, showing it is homeomorphic to a Hilbert space for smooth cases and maintains connectedness under finite-dimensional removals.

Findings

The space is homeomorphic to a separable Hilbert space when k is infinite.

Removing finite-dimensional subsets does not disconnect the space.

Similar connectedness results hold for the moduli space of metrics.

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.