Cao’s rigidity conjecture for complete shrinking gradient Ricci solitons

Determine whether every complete shrinking gradient Ricci soliton with constant scalar curvature is rigid, namely a finite quotient of a product N^k × R^{n−k}, where N^k is an Einstein manifold of positive scalar curvature.

Background

The paper studies complete shrinking gradient Ricci solitons with constant scalar curvature and proves rigidity under additional assumptions: an inequality involving the Ricci tensor outside a compact set and smooth convergence to R2 × S{n−2}. These hypotheses establish only a special case of the broader rigidity problem.

In the terminology recalled in the paper, a gradient Ricci soliton is rigid when it is isometric to a quotient of an Einstein manifold of positive scalar curvature times a Euclidean Gaussian factor. The unresolved conjecture asks whether constant scalar curvature alone forces this product structure in the complete shrinking case.

References

The question is therefore whether constant scalar curvature forces rigidity in the noncompact case.

— Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature  (2609.29359 - Horácio, 24 Sep 2026) in Section 1, Introduction and main results

They also observed that constant scalar curvature $R$ makes the second trace constant, whereas the third and fourth were ``not yet completely understood.''

— Rigidity of nonsteady gradient Kähler-Ricci solitons with constant scalar curvature  (2609.29359 - Horácio, 24 Sep 2026) in Section 1, Introduction and main results

Furthermore, for the complete shrinking case, Professor Huai-Dong Cao conjectured that $(Mn, g, f)$ has constant scalar curvature if and only if it is rigid, i.e., a finite quotient of ${N}k \times \mathbb{R}{n-k}$ for some Einstein manifold ${N}$ of positive scalar curvature.

— Rigidity of shrinking gradient ricci soliton with constant scalar curvature  (2608.20040 - Li et al., 20 Aug 2026) in Section 1, Introduction

Cao conjectured that every complete gradient shrinking Ricci soliton with constant scalar curvature $R$ is rigid (see ).

— On Ricci solitons whose level hypersurfaces have parallel second fundamental form  (2610.11209 - Horácio, 8 Oct 2026) in Section 1, Introduction and main results

For complete shrinkers, Cao conjectured that constant scalar curvature alone should imply rigidity.

— Rigidity of Kähler-Ricci Solitons with Constant Scalar Curvature  (2610.11935 - Horácio et al., 8 Oct 2026) in Section 1, Introduction