Semi-Continuity of the Morse Index for Ricci Shrinkers

TL;DR

This paper establishes semi-continuity properties of the Morse index for sequences of gradient Ricci shrinkers, including non-compact cases, using advanced analytical techniques and extending previous work on bubble tree convergence.

Contribution

It extends semi-continuity results of the Morse index to non-compact Ricci shrinkers and refines analytical methods for studying their convergence behavior.

Findings

Proved lower and upper semi-continuity of the Morse index for Ricci shrinkers.

Extended analysis to non-compact shrinkers with finite weighted volume.

Identified conditions linking the Morse index of shrinkers to their asymptotic cones.

Abstract

We prove lower and upper semi-continuity of the Morse index for sequences of gradient Ricci shrinkers which bubble tree converge in the sense of past work by the author and Buzano. Our proofs rely on adapting recent arguments of Workman which were used to study certain sequences of CMC hypersurfaces and were in turn adapted from work on Da Lio-Gianocca-Riviere. Moreover, we are able to refine Workman's methods by using techniques related to polynomially weighted Sobolev spaces. This all also requires us to extend the analysis to handle when the shrinkers we study are non-compact, which we can do due to the availability of a suitable notion of finite weighted volume. Finally, we identify a technical condition which ensures the Morse index of an asymptotically conical shrinker is bounded below by the f-index of its asymptotic cone.