This thesis studies canonical metrics on Riemannian manifolds. It consists of two parts. In the first part, we study Einstein four-manifolds with 3-positive curvature operator. We prove that if the curvature operator is 3-positive, then the sectional curvature has a positive lower bound which depends only on the Einstein constant; we also partially improve results of Brendle [Bre10], and Bohm-Wilking [BW08].

In the second part, we study gradient steady Ricci solitons. We first prove that the infimum of the potential function decays linearly, which implies that any gradient steady Ricci soliton with bounded potential function must be Ricci flat, and that no gradient steady Ricci soliton has uniformly positive scalar curvature. Further we show that if the potential function satisfies a uniform condition, then the gradient steady Ricci soliton has at most Euclidean volume growth.