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

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