Geometry Seminar: Reto Buzano (UniTo)
Geometry Seminar (Geometric Analysis)
Speaker: Reto Buzano (Università di Torino)
Title: Convergence of sequences of gradient shrinking Ricci solitons
Speaker: Reto Buzano (Università di Torino)
Title: Convergence of sequences of gradient shrinking Ricci solitons
Abstract: In 2011, in joint work with Robert Haslhofer, we proved that any sequence of gradient shrinking Ricci solitons with entropy uniformly bounded from below and uniform local energy bounds converges in a pointed orbifold-Cheeger-Gromov sense to an orbifold Ricci shrinker. In this talk, we will focus on two recent refinements of this compactness theory: Firstly, we will investigate what happens at the orbifold singularities that can form in the limit. We will show that in these points one gets full bubble-tree convergence, where the bubbles are Ricci-flat, finite energy ALE-orbifolds with precisely one end and we will prove an energy identity that shows that no energy is lost in “bubble-necks”. This analysis yields a local diffeomorphism finiteness result for gradient shrinking Ricci solitons under the above mentioned entropy and energy bounds. Secondly, we will study the number of ends along weakly converging sequences. Generally pointed convergence gives almost no control over what happens at infinity, so ends could in theory form, disappear, or merge in the limit. We will show that under a variety of assumptions, these phenomena cannot happen, for instance there can never be a new asymptotically conical end in the limit. The two results are joint work with Louis Yudowitz and Alessandro Bertellotti, respectively.