Bubble-tree convergence and local diffeomorphism finiteness for gradient Ricci shrinkers

Abstract We prove bubble-tree convergence of sequences gradient Ricci shrinkers with uniformly bounded entropy and uniform local energy bounds, refining the compactness theory Haslhofer Müller (Geom Funct Anal 21:1091–1116, 2011; Proc Am Math Soc 143(10):4433–4437, 2015). In particular, we show that no concentrates in neck regions, a result which implies identity for sequence. Direct consequences these results are an Euler characteristic diffeomorphism finiteness theorem.