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

TL;DR

This paper establishes bubble-tree convergence for sequences of gradient Ricci shrinkers with bounded entropy and energy, leading to new insights into their geometric structure and finiteness properties.

Contribution

It refines the compactness theory of Ricci shrinkers by proving bubble-tree convergence and a local energy identity, with implications for diffeomorphism finiteness.

Findings

No energy concentration in neck regions

Local energy identity established

Finiteness of local diffeomorphisms

Abstract

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