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Kähler Gradient Ricci Solitons with Large Symmetry

Hung Tran

TL;DR

The paper addresses the problem of classifying Kähler gradient Ricci solitons with large symmetry by linking symmetry to almost contact geometry on level sets of the soliton potential. It introduces a cohomogeneity-one Ansatz built from a Sasakian-type deformation over a base ${N}={ m N}(k)$ and shows that the isometry group is bounded by $n^2$, with equality characterized precisely by this construction. The results yield a unique $U(n)$-invariant family within this framework and yield corollaries for automorphism and affine transformation groups, including decompositions in irreducible vs. reducible cases. The approach combines Riemannian submersion techniques, almost contact metric geometry, and ODE reductions to establish rigidity and a complete picture of maximal-symmetry Kähler GRS. This advances understanding of the interplay between geometric structure, symmetry, and soliton equations, with potential implications for explicit soliton constructions and classification in higher dimensions.

Abstract

Let $(M, g, J, f)$ be an irreducible non-trivial Kähler gradient Ricci soliton of real dimension $2n$. We show that its group of isometries is of dimension at most $n^2$ and the case of equality is characterized. As a consequence, our framework shows the uniqueness of $U(n)$-invariant Kähler gradient Ricci solitons constructed earlier. There are corollaries regarding the groups of automorphisms or affine transformations and a general version for almost Hermitian GRS. The approach is based on a connection to the geometry of an almost contact metric structure.

Kähler Gradient Ricci Solitons with Large Symmetry

TL;DR

The paper addresses the problem of classifying Kähler gradient Ricci solitons with large symmetry by linking symmetry to almost contact geometry on level sets of the soliton potential. It introduces a cohomogeneity-one Ansatz built from a Sasakian-type deformation over a base and shows that the isometry group is bounded by , with equality characterized precisely by this construction. The results yield a unique -invariant family within this framework and yield corollaries for automorphism and affine transformation groups, including decompositions in irreducible vs. reducible cases. The approach combines Riemannian submersion techniques, almost contact metric geometry, and ODE reductions to establish rigidity and a complete picture of maximal-symmetry Kähler GRS. This advances understanding of the interplay between geometric structure, symmetry, and soliton equations, with potential implications for explicit soliton constructions and classification in higher dimensions.

Abstract

Let be an irreducible non-trivial Kähler gradient Ricci soliton of real dimension . We show that its group of isometries is of dimension at most and the case of equality is characterized. As a consequence, our framework shows the uniqueness of -invariant Kähler gradient Ricci solitons constructed earlier. There are corollaries regarding the groups of automorphisms or affine transformations and a general version for almost Hermitian GRS. The approach is based on a connection to the geometry of an almost contact metric structure.
Paper Structure (14 sections, 30 theorems, 112 equations)

This paper contains 14 sections, 30 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminaries
  4. Almost Complex Structure
  5. Gradient Ricci Solitons
  6. Group Actions on a manifold
  7. Almost Contact Structure
  8. Model Spaces
  9. Cohomogeneity One Ansatz
  10. Induced Symmetry
  11. Rigidity of the Maximal Dimension
  12. Appendix
  13. Convention
  14. Submersion

Key Result

Theorem 1.2

Let $(M^{n}, g, J, f, \lambda)$ be an irreducible non-trivial complete Kähler GRS. Its group of isometries is of dimension at most $n^2$ and equality happens iff it is smoothly constructed by the ansatz ansatz for ${N}=\mathbb{N}(k)$, $q=1$, and $f$ can be chosen as a function of $t$. If $\lambda\ge

Theorems & Definitions (76)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.4
  • Remark 1.5
  • ...and 66 more