Contractibility results for certain spaces of Riemannian metrics on the disc

The aim of this paper is to show that the space M+(D¯¯¯¯) of positively curved metrics on the 2-disk with mean convex boundary is contractible. This result can be anticipated from the fact that the space R+(S2) of positively curved metrics on the 2-sphere is also contractible, first proven in [J. M. Rosenberg and S. Stolz, in Surveys on surgery theory, Vol. 2, 353–386, Ann. of Math. Stud., 149, Princeton Univ. Press, Princeton, NJ, 2001; MR1818778]. The proof in this case uses the uniformization theorem to study the map π:M+(D¯¯¯¯)→C(D¯¯¯¯) onto the space of conformal classes of all metrics. The authors observe that the map Φ:C(D¯¯¯¯)→Diff(D¯¯¯¯) constructed in the various proofs of the uniformization theorem allows them to define a section σ:C(D¯¯¯¯)→M+(D¯¯¯¯) by σ([g]):=Φ([g])∗gD (for some base-point gD∈M+(D¯¯¯¯)). It is then an elementary observation from the curvature equations for conformal changes that the fibers of π are convex, and hence M+(D¯¯¯¯) deformation retracts onto the image of σ. Indeed, the main result is stated slightly more generally: any diffeomorphism invariant subspace of metrics for which the fibers of π are convex is contractible. This result then follows from the fact that C(D¯¯¯¯) is contractible.