Rigidity of Non-Steady Gradient Ricci Solitons

Let (M,g) be a connected, compact Riemannian manifold of dimensionan n. We demonstrate that, after a suitable normalization, a shrinking gradient Ricci soliton (M,g,f,λ) is trivial exactly when the mean value of f is less than or equal to n2. Moreover, we prove that a normalized non-steady gradient Ricci soliton (M,g,f,λ) is trivial if and only if its scalar curvature S satisfies the relation S=λf+n2. In addition, we establish that if (M,g,f,λ) admits an isometric immersion as a hypersurface in the Euclidean space, then the soliton must necessarily be of a shrinking type. In such a case, the constant λ and the mean curvature of M satisfy a certain inequality, with equality occurring precisely when M is isometric to a round sphere.