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.

Aspects of Positive Scalar Curvature and Topology II

“…A more recent example can be found in [18]. We should also point out that there are a number of interesting results concerning topological non-triviality in the moduli space of non-negative sectional curvature metrics for certain open manifolds; see in particular work by Belegradek, Kwasik and Schultz [5], Belegradek and Hu [4] and very recently Belegradek, Farrell and Kapovitch [3].…”

Spaces of nonnegatively curved surfaces

“…is a certain diffeomorphism group and O γ ≥0 (C) is some convex set of smooth functions on C. The homeomorphism type of the factors D γ+1 (C), O γ ≥0 (C) can be determined along the lines of the proof of Theorem 1.1, and the conclusion is that D γ+1 (C), O γ ≥0 (C) are each homeomorphic to ℓ 2 or Σ ω depending on whether γ is infinite or finite. The case R γ >0 (C) is similar, and the proof of Theorem 1.2 follows the same outline except that [BH15] is not needed. The above proof requires that all metrics lie in the same conformal class, and in particular, the proof does not extend to R γ ≥λ (C) with λ < 0 or to R γ ≥λ (M ) where M is a closed surface of nonpositive Euler characteristic.…”

“…The homeomorphism of R ∞ ≥0 (C) and ℓ 2 was already established in [BH15]. To explain the assumption γ / ∈ Z, let us sketch the analytic ingredients of Theorem 1.3 in the case of R γ ≥0 (C).…”

“…Hence it can be written as φ * e −2u g 0 where φ is an orientation-preserving diffeomorphism of C and u is a smooth function on C. By composing φ with a conformal automorphism of C one can assume that φ fixes a pair of points, say 0 and 1, which determines φ uniquely. Such φ's form a closed contractible subgroup of the diffeomorphism group of C, which we denote D γ+1 (C); the superscript γ + 1 indicates that the group is equipped with the C γ+1 topology, which is natural because φ * e −2u g 0 involves the differential of φ. Nonnegativity of the curvature is equivalent to subharmonicity of u. Completeness of e −2u g 0 imposes further restrictions on u, making it roughly speaking of sublogorithmic growth, and such u's form a convex subset O γ ≥0 (C) in the space of smooth functions equipped with the C γ topology, see [BH15]. With the above restrictions on φ and u the map (φ, u) → φ * e −2u g 0 becomes a continuous bijection.…”

“…With the above restrictions on φ and u the map (φ, u) → φ * e −2u g 0 becomes a continuous bijection. Moreover, smooth dependence of solutions of Beltrami equation on the dilatation shows that the continuous bijection is a homeomorphism for γ / ∈ Z, see [BH15,BH16]; we expect this to fail when γ is an integer. In summary, for γ /…”