The time-dependent maximum principle for systems of parabolic equations   subject to an avoidance set
  In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to an avoidance set. 
Bennett Chow and Peng Lu