Nonminimal De Rham-Hodge Operators and Non-commutative Residue
  In this paper, we get a Kastler-Kalau-Walze type theorem associated to nonminimal de Rham-Hodge operators on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of four dimensional compact manifolds with flat boundary. 
Jian Wang, Yong Wang, Aihui Sun, Sihui Chen
math.DG