Analysis and Mathematical Physics

Ricci solitons, introduced by R. Hamilton in the mid-80s, are self-similar solutions to the Ricci flow and natural generalizations of Einstein manifolds. Shrinking Ricci solitons, in particular,  model Type I singularities of the Ricci flow and arise as critical points of Perelman's $\nu$-entropy. In this talk, after a quick introduction on (gradient) Ricci solitons, we shall discuss the linear stability of shrinking Ricci solitons with respect to Perelman's \nu-entropy and the first eigenvalue estimates of the Laplace–Beltrami/Lichnerowicz-type Laplacian operators.