We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci–DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing the mean curvature and conformal class of the boundary, with arbitrary initial data. Finally, we establish that under suitable control of the boundary data the flow exists as long as the ambient curvature and the second fundamental form of the boundary remain bounded.
"The Ricci flow on manifolds with boundary." J. Differential Geom. 104 (2) 291 - 324, October 2016. https://doi.org/10.4310/jdg/1476367059