Remarks on complete non-compact gradient Ricci expanding solitons

Abstract

In this paper, we study gradient Ricci expanding solitons (X,g) satisfying

Rc = cg + D²f,

where Rc is the Ricci curvature, c < 0 is a constant, and D²f is the Hessian of the potential function f on X. We show that for a gradient expanding soliton (X,g) with non-negative Ricci curvature, the scalar curvature R has at most one maximum point on X, which is the only minimum point of the potential function f. Furthermore, R > 0 on X unless (X,g) is Ricci flat. We also show that there is exponentially decay for scalar curvature on a complete non-compact expanding soliton with its Ricci curvature being ε-pinched.