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We prove bubble-tree convergence of sequences of gradient Ricci shrinkerswith uniformly bounded entropy and uniform local energy bounds, refining thecompactness theory of Haslhofer-Mueller. In particular, we show that no energyconcentrates in neck regions, a result which implies a local energy identityfor the sequence. Direct consequences of these results are an identity for theEuler characteristic and a local diffeomorphism finiteness theorem.

Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers