  In this paper, we study the Ricci flow on manifolds with boundary. In the
first part of the paper, we prove short-time existence and uniqueness of the
solution, in which the boundary becomes instantaneously umbilic for positive
time. In the second part of the paper, we prove that the flow we constructed in
the first part preserves natural boundary conditions. More specifically, if the
initial metric has a convex boundary, then the flow preserves positive
curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial
metric has a two-convex boundary, then the flow preserves the PIC condition.
