Contractibility results for certain spaces of Riemannian metrics on the disc

We provide a general contractibility criterion for subsets of Riemannian metrics on the disc. For instance, this result applies to the space of metrics that have positive Gauss curvature and make the boundary circle convex (or geodesic). The same conclusion is not known in any dimension $n \geq 3$, and (by analogy with the closed case) is actually expected to be false for many values of $n \geq 4$.

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A.C. is partly supported by the National Science Foundation (through grant DMS 1638352) and by the Giorgio and Elena Petronio Fellowship Fund. D.W. is partly supported by the National Science Foundation (through grant DMS 1611745) and the Simonyi Endowment of the Institute for Advanced Study.

Received 21 August 2019

Accepted 23 December 2019

Published 22 November 2021