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Boundary estimates for the Ricci flow

In this paper we consider the Ricci flow on manifolds with boundary with appropriate control on its mean curvature and conformal class. We obtain higher order estimates for the curvature and second fundamental form near the boundary, similar to Shi's local derivative estimates. As an application, we prove a version of Hamilton's compactness theorem in which the limit has boundary. Finally, we show that in dimension three the second fundamental form of the boundary and its derivatives are a priori controlled in terms of the ambient curvature and some non-collapsing assumptions. In particular, the flow exists as long as the curvature remains bounded, in contrast to the general case where control on the second fundamental form is also required.