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 of Da Lio-Gianocca-Rivière. 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.
[Arχiv]
We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics.
[Arχiv]
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-Müller. 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.
[Arχiv, Journal]
In this article, we prove a general and rather flexible upper bound for the heat kernel of a weighted heat operator on a closed manifold evolving by an intrinsic geometric flow. The proof is based on logarithmic Sobolev inequalities and ultracontractivity estimates for the weighted operator along the flow, a method which was previously used by Davies in the case of a non-evolving manifold. This result directly implies Gaussian-type upper bounds for the heat kernel under certain bounds on the evolving distance function; in particular we find new proofs of Gaussian heat kernel bounds on manifolds evolving by Ricci flow with bounded curvature or positive Ricci curvature. We also obtain similar heat kernel bounds for a class of other geometric flows.
[Arχiv, Journal]