A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Abstract: Let $(M^n, g, f)$ be an $n$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. 1. If its scalar curvature is $\frac{k}{2}$, Ricci curvature is nonnegative and sectional curvature has upper bound $\frac{1}{2(k-1)}$, we prove that the Ricci shrinker is isometric to a finite quotient of $\mathbb{R}^{n-k}\times \mathbb{S}^k$. 2. If $M$ has constant scalar curvature $R=\frac{n-2}{2}$, and each level set of $f$ has vanishing Weyl curvature, we prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^{n-2}$. This can be seen a generalization of Cheng-Zhou's four dimensional result \cite{Cheng-Zhou} to high dimension, since the level set of the potential function $f$ has vanishing Weyl curvature automatically when $n=4$.

• The paper establishes two main rigidity theorems for complete noncompact shrinking Ricci solitons with constant scalar curvature.
• It employs weighted Laplacian and Ricci eigenvalue estimates to derive key curvature inequalities and ensure nonnegative Ricci curvature.
• Findings generalize rigidity in higher dimensions under vanishing Weyl curvature and sectional curvature bounds, refining previous classifications.

Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

Introduction and Background

The paper "A note on Rigidity of Shrinking Gradient Ricci Solitons with Constant Scalar Curvature" (2604.23939) investigates the rigidity phenomenon in the classification of complete noncompact shrinking gradient Ricci solitons, specifically for those possessing constant scalar curvature. Gradient Ricci solitons, defined by the equation $\mathrm{Ric} + \nabla^2 f = \lambda g$, serve as self-similar solutions to the Ricci flow and play a central role in the analysis of geometric flows and singularities. The canonical normalization $\lambda = \frac{1}{2}$ is used throughout, targeting shrinking solitons.

A major conjecture, attributed to Huai-Dong Cao, postulates that every complete $n$-dimensional gradient shrinking Ricci soliton with constant scalar curvature ($n \geq 4$) must be rigid, i.e., isometric to a finite quotient of $\mathbb{N}^k \times \mathbb{R}^{n-k}$ for some Einstein manifold $\mathbb{N}$ of positive scalar curvature. This conjecture was previously confirmed only in dimension four [Cheng-Zhou].

Main Results

The authors address two principal rigidity theorems:

1. Rigidity under Vanishing Weyl Curvature in Level Sets: If $(M^n, g, f)$ is a complete shrinking Ricci soliton with constant scalar curvature $R = \frac{n-2}{2}$ and each level set of $f$ has vanishing Weyl curvature, then $M$ is isometric to a finite quotient of $\lambda = \frac{1}{2}$0.
2. Rigidity with Sectional Curvature Upper Bound: If $\lambda = \frac{1}{2}$1 is a complete shrinking Ricci soliton with constant scalar curvature $\lambda = \frac{1}{2}$2, nonnegative Ricci curvature, and sectional curvature bounded above by $\lambda = \frac{1}{2}$3, then $\lambda = \frac{1}{2}$4 is isometric to a finite quotient of $\lambda = \frac{1}{2}$5.

Both theorems generalize prior results in dimension four and five to arbitrary higher dimensions. The proofs synthesize weighted Laplacian estimates on Ricci eigenvalues and sharp curvature analysis.

Technical Approach

Weighted Laplacian and Ricci Eigenvalue Estimates

The central analytical tool is the weighted Laplacian (Bakry-Émery), $\lambda = \frac{1}{2}$6, applied to sums of minimal Ricci eigenvalues. For solitons with scalar curvature $\lambda = \frac{1}{2}$7, it is proved that Ricci curvature is nonnegative and the smallest Ricci eigenvalue vanishes, aligning with the eigenvector $\lambda = \frac{1}{2}$8. The sum $\lambda = \frac{1}{2}$9 is shown to satisfy a differential inequality involving sectional curvatures and Weyl curvature contributions from level sets of $n$0.

The key inequality derived,

$n$1

(where the ellipsis denotes explicit positive lower order terms), serves as the cornerstone for analyzing rigidity by induction, maximum principle, and distributional integration.

Curvature Boundedness and Point-Picking Arguments

Using pseudolocality and compactness results under Ricci flow, along with $n$2-noncollapsed volume estimates, the curvature of the soliton is demonstrated to be bounded. A contradiction is constructed in the hypothetical case of unbounded curvature, ultimately leading to a Cheeger-Gromoll splitting and showing nonflatness of the limit, incompatible with vanishing Weyl curvature.

Classification of Level Sets and Splitting

The soliton's potential function $n$3 is shown to be isoparametric; level sets at infinity are classified using the Gauss equation and analysis of the Weyl tensor. Under vanishing Weyl curvature assumptions, level sets in the limit split as $n$4 spheres, and the soliton globally splits as $n$5 (or higher-dimensional analogues), confirming rigidity.

Sectional Curvature Bound and Rank Arguments

For the theorem involving sectional curvature upper bounds, upper bounds enforce constant rank properties for the Ricci tensor. Application of the strong maximum principle, gradients estimates, and De Rham splitting yield a direct product structure with the Einstein manifold being the sphere of maximal allowed sectional curvature.

Numerical and Structural Implications

Both rigidity theorems are sharp. The upper sectional curvature bound matches the constant curvature of spheres, and the rank splitting corresponds to the eigenvalue multiplicity of the Ricci tensor. The arguments exclude nontrivial conformally flat, noncompact cases in higher dimensions; only cylinders and flat products persist under rigidity, which is consistent with results from Caffarelli-Gidas-Spruck and Zhu.

Impact and Future Directions

The analytic classification achieved extends and completes previously partial results in the field. The reliance on eigenvalue methods and weighted Laplacian inequalities offers a blueprint for further generalizations, such as solitons with weaker curvature or nontrivial Weyl fields. Future developments may explore rigidity of expanding or steady solitons, variational approaches under metric measure structures, and implications for singularity formation in geometric flows.

The results constrain possible noncompact shrinking solitons with constant scalar curvature, further simplifying the landscape for Ricci flow singularity analysis and soliton moduli space classification. Potential theoretical extensions include rigidity for Kähler-Ricci solitons and refinements in metric measure geometry.

Conclusion

This paper rigorously establishes rigidity theorems for shrinking gradient Ricci solitons with constant scalar curvature under vanishing Weyl curvature of level sets or sharp sectional curvature bounds. By employing analytic methods on weighted Laplacians and eigenvalues, along with geometric compactness arguments and splitting theorems, it generalizes prior rigidity results to all dimensions. The findings decisively classify solitons under these constraints, underpinning both the geometric and analytic structure of Ricci flows and soliton solutions.