Classification and geometry of higher-dimensional gradient shrinking Ricci solitons

Classify the general gradient shrinking Ricci solitons in dimensions four and higher and determine their geometric structure, extending the complete classifications known in dimensions two and three.

Background

Complete gradient shrinking Ricci solitons have been fully classified in dimensions two and three, whereas the paper identifies the corresponding classification and geometric understanding in dimensions four and higher as an unresolved problem. The almost-Kähler hypothesis studied in the paper provides one structured class for which the geometry reduces to that of Kähler–Ricci shrinkers, but it does not resolve the general higher-dimensional problem.

References

By contrast, the classification and geometry of general gradient shrinking Ricci solitons in dimensions four and higher remain largely open; see for surveys of the subject and further references.

— Kählerity of complete almost-Kähler gradient shrinking Ricci solitons  (2609.00840 - Xie, 1 Sep 2026) in Section 1, Introduction

In dimension four and higher, the classification problem remains largely open, yet substantial progress has been achieved by imposing various curvature pinching restrictions; see, e.g., and the references therein.

— Four-dimensional shrinkers with pinched curvatures  (2610.11040 - Cao et al., 8 Oct 2026) in Section 1, Introduction

Since shrinkers are generalizations of Einstein manifolds, and non-trivial four-dimensional shrinkers on connected sums $\mathbb{CP}2\sharp\, k(-\mathbb{CP}2)$ $(k=1,2)$ satisfy the classical Hitchin-Thorpe inequality, H.-D. Cao raised a natural question: does an analogous Hitchin-Thorpe type inequality hold for arbitrary closed four-dimensional shrinkers?

— Four-dimensional shrinkers with pinched curvatures  (2610.11040 - Cao et al., 8 Oct 2026) in Section 1, Introduction

But to the authors, these equalties seem not to imply $=0$.

— Four-dimensional shrinkers with pinched curvatures  (2610.11040 - Cao et al., 8 Oct 2026) in Remark following the proof of Theorem 1.1, in Section 3.1