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Local structure of gradient almost Ricci solitons with harmonic Weyl tensor

Valter Borges, Matheus Andrade Ribeiro de Moura Horácio, João Paulo dos Santos

TL;DR

This work analyzes gradient almost Ricci solitons with harmonic Weyl curvature in dimensions n ≥ 4, establishing that the Ricci tensor can have at most three distinct eigenvalues near regular points of the potential function f. It then shows that such solitons are locally realizable as multiply warped products with a one-dimensional base and at most two Einstein fibers, and it derives precise relations among the warping functions, f, and the soliton parameter λ. In the two-eigenvalue case, the authors provide a complete local classification: either the manifold is a warped product with an Einstein fiber and explicit f, λ, or it splits as a product with a Ricci-flat base and a fiber with constant curvature under completeness; a rigidity result further describes complete nonradially flat cases as rigid solitons on a Euclidean base. The results extend prior work by Catino and Kim, offering a broader dimensional perspective and a polynomial-based argument to bound the number of fibers, yielding a detailed structural and analytical picture of these solitons.

Abstract

In this article, we investigate a gradient almost Ricci soliton with harmonic Weyl tensor. We first prove that its Ricci tensor has at most three distinct eigenvalues of constant multiplicities in a neighborhood of a regular point of the potential function. Then, we classify those with exactly two distinct eigenvalues. It is worth mentioning that the case with exactly one eigenvalue has already been settled elsewhere. Our results are based on a local representation of these manifolds as multiply warped products of a one-dimensional base, having at most two Einstein fibers, which we also obtain in this paper. These results extend a result by Catino, who assumes, in addition, that the Weyl tensor is radially flat, and a result by Kim, who considers the four-dimensional case.

Local structure of gradient almost Ricci solitons with harmonic Weyl tensor

TL;DR

This work analyzes gradient almost Ricci solitons with harmonic Weyl curvature in dimensions n ≥ 4, establishing that the Ricci tensor can have at most three distinct eigenvalues near regular points of the potential function f. It then shows that such solitons are locally realizable as multiply warped products with a one-dimensional base and at most two Einstein fibers, and it derives precise relations among the warping functions, f, and the soliton parameter λ. In the two-eigenvalue case, the authors provide a complete local classification: either the manifold is a warped product with an Einstein fiber and explicit f, λ, or it splits as a product with a Ricci-flat base and a fiber with constant curvature under completeness; a rigidity result further describes complete nonradially flat cases as rigid solitons on a Euclidean base. The results extend prior work by Catino and Kim, offering a broader dimensional perspective and a polynomial-based argument to bound the number of fibers, yielding a detailed structural and analytical picture of these solitons.

Abstract

In this article, we investigate a gradient almost Ricci soliton with harmonic Weyl tensor. We first prove that its Ricci tensor has at most three distinct eigenvalues of constant multiplicities in a neighborhood of a regular point of the potential function. Then, we classify those with exactly two distinct eigenvalues. It is worth mentioning that the case with exactly one eigenvalue has already been settled elsewhere. Our results are based on a local representation of these manifolds as multiply warped products of a one-dimensional base, having at most two Einstein fibers, which we also obtain in this paper. These results extend a result by Catino, who assumes, in addition, that the Weyl tensor is radially flat, and a result by Kim, who considers the four-dimensional case.
Paper Structure (13 sections, 36 theorems, 102 equations)

This paper contains 13 sections, 36 theorems, 102 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Results for when the Weyl tensor is harmonic
  4. Multiply warped products
  5. Gradient almost solitons whose Cotton tensor vanishes
  6. $R$, $\lambda$ and the eigenvalues of $\mathrm{Ric}$ depend only on $s$
  7. Local multiply warped product structure
  8. The almost Ricci soliton equations with at most two fibers
  9. Equations when there is only one fiber
  10. Equations when there are two fibers
  11. Harmonic Weyl tensor: exactly two eigenvalues
  12. Appendix: functions satisfying \ref{["sit_f_1'"]} and \ref{["sit_f_2'"]} are constant
  13. Acknowledgments

Key Result

Theorem 1.1

Let $(M^n,g,f,\lambda)$, $n\geq4$, be a gradient almost Ricci soliton with harmonic Weyl curvature with $f$ nonconstant. Then the Ricci tensor of $M$ has at most three distinct eigenvalues at each point of $M$. If $p\in M$ is a regular point of $f$, then there is an open neighborhood of $p$ such tha

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Lemma 2.1: Lemma 2.1 of kim1
  • Lemma 2.2: Lemma 2.2 of kim1
  • Lemma 2.3: Derdziński
  • Remark 2.4: Counting fibers
  • ...and 49 more