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Backward propagation of warped product structures and asymptotically conical shrinkers

Brett Kotschwar

TL;DR

The paper develops a backward-Ricci-flow framework to study how warped-product structures propagate backward in time. By introducing time-dependent horizontal/vertical distributions and a system of invariants (including $A$, $T$, $M$, $P$, and $U$) that quantify deviations from warped-product geometry, the authors derive closed PDE-ODE inequalities and apply a backward-uniqueness principle to conclude rigidity results. A central application is to asymptotically conical shrinking solitons: if the end is asymptotic to a cone with cross-section a product of Einstein manifolds, the shrinker must be a multiply warped product with the same base and fiber factors, determined by the cone. This yields structural rigidity, linking the soliton to its asymptotic cone and enabling differentiation of isometry groups and Kähler properties in AC shrinkers. Overall, the work provides a robust method to extract global geometric structure from asymptotic cone data via backward propagation under Ricci flow.

Abstract

We establish sufficient conditions which ensure that a locally-warped product structure propagates backward in time under the Ricci flow. As an application, we prove that if an asymptotically conical gradient shrinking soliton is asymptotic to a cone whose cross-section is a product of Einstein manifolds, the soliton must itself be a multiply-warped product over the same manifolds.

Backward propagation of warped product structures and asymptotically conical shrinkers

TL;DR

The paper develops a backward-Ricci-flow framework to study how warped-product structures propagate backward in time. By introducing time-dependent horizontal/vertical distributions and a system of invariants (including , , , , and ) that quantify deviations from warped-product geometry, the authors derive closed PDE-ODE inequalities and apply a backward-uniqueness principle to conclude rigidity results. A central application is to asymptotically conical shrinking solitons: if the end is asymptotic to a cone with cross-section a product of Einstein manifolds, the shrinker must be a multiply warped product with the same base and fiber factors, determined by the cone. This yields structural rigidity, linking the soliton to its asymptotic cone and enabling differentiation of isometry groups and Kähler properties in AC shrinkers. Overall, the work provides a robust method to extract global geometric structure from asymptotic cone data via backward propagation under Ricci flow.

Abstract

We establish sufficient conditions which ensure that a locally-warped product structure propagates backward in time under the Ricci flow. As an application, we prove that if an asymptotically conical gradient shrinking soliton is asymptotic to a cone whose cross-section is a product of Einstein manifolds, the soliton must itself be a multiply-warped product over the same manifolds.
Paper Structure (40 sections, 34 theorems, 390 equations)

This paper contains 40 sections, 34 theorems, 390 equations.

Table of Contents

  1. Introduction
  2. Forward propagation of warped-product structures
  3. Backward propagation
  4. Asymptotically conical shrinking solitons
  5. The determination of the shrinker from its cone
  6. Connection invariants associated to complementary orthogonal distributions
  7. Invariants associated to a pair of complementary orthogonal distributions
  8. A remark on the notation
  9. A characterization of locally-warped products.
  10. Some notational conventions.
  11. Barred and underlined indices
  12. Contraction with the mean-curvature vector $N$
  13. Notation for covariant derivatives
  14. Asterisk notation
  15. Other conventions
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $(B, \breve{g})$ and $(F, \bar{g})$ be Riemannian manifolds, where $\bar{g}$ is Einstein. Suppose $g(\tau)$ is a smooth solution to eq:brf on $M = B\times F$ for $\tau\in [0, \Omega]$ which satisfies for some positive $h_{0}\in C^{\infty}(B)$. Assume that $\sup_{M\times [0, \Omega]}|\operatorname{Rm}| <\infty$. and, if $B$ is noncompact, that $\sup_{B}|\breve{\nabla}\log h_{0}| < \infty$. The

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 2.1
  • Lemma 2.2: cf. Besse, Proposition 9.104
  • proof
  • Lemma 2.5
  • proof
  • ...and 56 more