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Semi-Continuity of the Morse Index for Ricci Shrinkers

Louis Yudowitz

TL;DR

This work establishes both upper and lower semi-continuity for the f-Morse index of gradient Ricci shrinkers under bubble-tree convergence to orbifold shrinkers, with a precise inequality Ind_f(M_∞) + ∑_{q∈𝒬} ∑_{k=1}^{N_q} Ind(V^k) ≤ liminf_{i→∞} Ind_f(M_i) and a complementary bound incorporating nullities. The analysis extends to non-compact shrinkers by leveraging a finite weighted volume and introduces a weighted spectral framework, using Lorentz spaces to manage blow-up near orbifold points and neck regions. A key technical achievement is proving that weighted and unweighted eigenspaces have the same dimension on body and bubble regions, while neck regions contribute no index, enabling a clean decomposition along the bubble tree. An additional cone-bound condition shows the f-index of an asymptotically conical end is controlled from below by the f-index of its asymptotic cone, connecting end geometry to stability. Together, these results yield a robust semi-continuity theory for the linear stability of shrinkers under orbifold convergence with potential applications to compactness and structural stability in Ricci flow singularity analysis.

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.

Semi-Continuity of the Morse Index for Ricci Shrinkers

TL;DR

This work establishes both upper and lower semi-continuity for the f-Morse index of gradient Ricci shrinkers under bubble-tree convergence to orbifold shrinkers, with a precise inequality Ind_f(M_∞) + ∑_{q∈𝒬} ∑_{k=1}^{N_q} Ind(V^k) ≤ liminf_{i→∞} Ind_f(M_i) and a complementary bound incorporating nullities. The analysis extends to non-compact shrinkers by leveraging a finite weighted volume and introduces a weighted spectral framework, using Lorentz spaces to manage blow-up near orbifold points and neck regions. A key technical achievement is proving that weighted and unweighted eigenspaces have the same dimension on body and bubble regions, while neck regions contribute no index, enabling a clean decomposition along the bubble tree. An additional cone-bound condition shows the f-index of an asymptotically conical end is controlled from below by the f-index of its asymptotic cone, connecting end geometry to stability. Together, these results yield a robust semi-continuity theory for the linear stability of shrinkers under orbifold convergence with potential applications to compactness and structural stability in Ricci flow singularity analysis.

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.
Paper Structure (16 sections, 23 theorems, 151 equations)

This paper contains 16 sections, 23 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Identities and Results for Gradient Shrinkers
  4. Bubble Tree Convergence
  5. Properties of the Stability Operators
  6. The Drift Einstein Operator on Shrinkers
  7. The Einstein Operator on the Bubbles
  8. Upper Semi-Continuity
  9. Lorentz Spaces
  10. The Weighted Eigenvalue Problem
  11. Body Region Analysis
  12. Bubble Region Analysis
  13. Neck Region Analysis
  14. Equivalence of the Weighted and Unweighted Eigenspaces
  15. Proof of the Upper Semi-Continuity Estimate
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $n \geq 4$ and let $\left(M_i,g_i,f_i\right)$ be a sequence of $n$-dimensional gradient Ricci shrinkers with entropy uniformly bounded below $\mu\left(g_i\right)\geq \underline{\mu}>-\infty$ which are normalized such that $\left(4\pi\right)^{-\frac{n}{2}}\int_M e^{-f} dV_g = 1$. If $n \geq 5$, t Then, with $p_i$ a minimum of the potential function $f_i$, after passing to a subsequence $\left(M

Theorems & Definitions (53)

  • Theorem 1.2
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.8
  • Definition 1.9
  • Theorem 1.10
  • Lemma 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • ...and 43 more