Shrinking Gradient Ricci Solitons

• Shrinking gradient Ricci solitons are complete Riemannian manifolds with a potential function f that satisfies the equation Ric + ∇²f = λg, serving as models for type I singularities in Ricci flow.
• They exhibit strong rigidity under supplementary curvature conditions such as Bach-flatness and conformal symmetry, leading to precise classifications in specific dimensions.
• Analytical methods involving tensor identities and maximum principles reveal splitting behaviors and asymptotic geometry, crucial for understanding their singularity formation and classification.

A shrinking gradient Ricci soliton is a complete Riemannian manifold $(M, g)$ together with a smooth function $f$ (the potential) satisfying $\mathrm{Ric} + \nabla^2 f = \lambda g$ for some constant $\lambda > 0$. Such solitons are self-similar solutions to the Ricci flow and model the formation of Type I singularities. Shrinking gradient Ricci solitons (often abbreviated as "shrinkers") are structurally rigid under various supplementary curvature conditions, and their precise classification in particular geometric regimes is a central topic in geometric analysis and the study of Ricci flow singularity models.

1. Definition and Canonical Examples

A gradient shrinking Ricci soliton is defined by the equation

$\mathrm{Ric} + \nabla^2 f = \lambda g$

where $\mathrm{Ric}$ is the Ricci tensor, $\nabla^2 f$ denotes the Hessian of the potential $f$, and $\lambda>0$ (commonly normalized to $\lambda = \frac12$). The canonical example is the Gaussian shrinker on $(\mathbb{R}^n, g_{\mathrm{Eucl}})$, with $f(x) = \frac{|x|^2}{4}$: $$\mathrm{Ric} = 0,\quad \nabla^2 f = \frac12 g,\quad \text{hence} \quad \mathrm{Ric} + \nabla^2 f = \frac12 g.$$ Other classical models include round spheres, product metrics such as $S^k \times \mathbb{R}^{n-k}$, and quotients thereof.

2. Rigidity via Curvature Conditions and Conformal Geometry

Rigidity of shrinking gradient Ricci solitons is sharply enhanced by additional curvature requirements. For instance, if the manifold is Bach-flat (i.e., its Bach tensor vanishes), classification in dimension $n \geq 4$ is as follows (Cao et al., 2011):

• In dimension $4$, the only possibilities are Einstein solitons or locally conformally flat solitons. Locally conformally flat gradient shrinking solitons are explicitly finite quotients of the Gaussian shrinker $\mathbb{R}^4$ and the round cylinder $S^3 \times \mathbb{R}$.
• In dimensions $n \geq 5$, any complete Bach-flat shrinker is either Einstein, a finite quotient of the Gaussian soliton $\mathbb{R}^n$, or a product $N^{n-1}\times \mathbb{R}$ with $N^{n-1}$ Einstein and positive scalar curvature.

This rigidity arises via the vanishing of specialized curvature tensors (Bach, Weyl, Cotton, etc.). For example, the complete vanishing of the three-tensor $D_{ijk}$ (which, in four dimensions, is equivalent to Bach-flatness) forces total umbilicity of level sets of $f$ and, depending on the dimension, the vanishing or harmonicity of the Weyl tensor. In dimension $4$, the result is that non-Einstein shrinkers must be locally conformally flat; in higher dimensions, the harmonicity of the Weyl tensor combined with the Ricci soliton structure yields a splitting.

Table: Rigidity consequences for shrinking gradient Ricci solitons under different curvature conditions

Curvature Condition	Dimension 4 Outcome	Higher Dimensions Outcome
Bach-flat	Einstein or locally conformally flat<br>($\mathbb{R}^4$, $S^3\times\mathbb{R}$ quotients)	Einstein, or $\mathbb{R}^n$, or $N^{n-1}\times\mathbb{R}$ quotient with $N^{n-1}$ Einstein
Positive isotropic curvature	$S^4$ quotient or $S^3\times\mathbb{R}$ quotient (Li et al., 2016)	—
Half harmonic Weyl	Einstein, $S^3 \times \mathbb{R}$, $S^2 \times \mathbb{R}^2$, or $\mathbb{R}^4$ (all up to finite quotients) (Wu et al., 2014)	—

3. Analytical Methods: Tensor Identities and Maximum Principles

Central to these rigidity results is the systematic use of tensorial identities involving the potential $f$, level set geometry, and curvature decomposition. For example, in dimension $4$ the D-tensor can be related to the Cotton and Weyl tensors by

$D_{ijk} = C_{ijk} + W_{ijkl}\,\nabla_l f,$

and for Bach-flat metrics, the vanishing of $B_{ij} \implies D_{ijk}\equiv 0$, forcing rigidity of the level set geometry of $f$. Further identities connect the norm of $D$ to the extrinsic geometry of level sets, e.g.

$$|D|^2 = C \big( |h|^2 - \frac{H^2}{n-1} \big) |\nabla f|^4,$$

where $h$ is the second fundamental form and $H$ its mean curvature.

Maximum principles for elliptic operators with respect to the weighted Laplacian $$\Delta_f = \Delta - \langle \nabla f, \cdot \rangle$$ are utilized to conclude vanishing of certain curvature quantities and hence to force splitting or constancy results (Ou et al., 10 Nov 2024). In the classification of solitons with harmonic half Weyl tensor (Wu et al., 2014), the maximum principle is decisive once a suitable Weitzenböck formula is derived.

4. Classification in Four Dimensions

Four-dimensional shrinking Ricci solitons exhibit particularly strong classification statements under various geometric conditions:

• Bach-flat shrinkers: Only Einstein metrics, or finite quotients of $\mathbb{R}^4$ or $S^3 \times \mathbb{R}$ (Cao et al., 2011).
• Positive isotropic curvature: Only $\mathbb{S}^4$ or $S^3\times \mathbb{R}$ quotients (Li et al., 2016).
• Half harmonic Weyl curvature ($\delta W^{\pm}=0$): Only Einstein, $S^3 \times \mathbb{R}$, $S^2 \times \mathbb{R}^2$, or $\mathbb{R}^4$ up to finite quotients (Wu et al., 2014).
• Pinching conditions on the (anti)self-dual Weyl tensor: Only Einstein, $\mathbb{R}^4$, $S^3 \times \mathbb{R}$, or $S^2 \times \mathbb{R}^2$ (Cao et al., 2020).
• Constant scalar curvature $R=1$: Isometric to a finite quotient of $\mathbb{R}^2\times S^2$ (Ou et al., 10 Nov 2024).

These statements are substantiated using decomposition of the curvature tensor (notably the splitting of 2-forms into self-dual and anti-self-dual parts), analysis of curvature eigenvalues, and fine tensorial inequalities. Integrability at infinity or volume growth conditions further restrict possible geometries to round spheres or their quotients (Shaikh et al., 2020).

5. Asymptotics and Curvature Control

For noncompact solitons, the geometry at infinity is tightly controlled if the scalar curvature decays. In four dimensions:

• The norm of the Riemann curvature tensor $|\mathrm{Rm}|$ is universally controlled by the scalar curvature $S$: $|\mathrm{Rm}| \leq c\,S$, and $|\mathrm{Rm}|$ is asymptotically nonnegative at infinity (Munteanu et al., 2014).
• If $S(x) \to 0$ as $x \to \infty$, then the soliton is asymptotically conical with $C^k$ asymptotics for all $k$.
• For Kähler-Ricci solitons whose scalar curvature vanishes at infinity and have bounded Ricci curvature, only the flat soliton on $\mathbb{C}^n$ or U(n)-invariant Feldman–Ilmanen–Knopf solitons on line bundles over $\mathbb{P}^{n-1}$ are possible (Conlon et al., 2019).
• In dimension $2$, all solitons with scalar curvature tending to zero at infinity are, up to pullback by $GL(2,\mathbb{C})$, either the flat Gaussian on $\mathbb{C}^2$ or the Feldman–Ilmanen–Knopf example on the one-point blowup of $\mathbb{C}^2$.

In general, the decay of curvature and suitable volume lower bounds (including via f-volume or Perelman's reduced volume) are essential in gap and rigidity theorems (Zhang, 2019).

6. Solitons with Symmetry: Kähler and Sasaki-Ricci Cases

Shrinking Kähler-Ricci solitons and their Sasaki analogues admit further structural features:

• If a shrinking gradient Ricci soliton is asymptotic to a Kähler cone at infinity, then it is actually Kähler near infinity and, if complete, globally Kähler (Kotschwar, 2017).
• On noncompact toric manifolds, up to automorphisms, there is at most one $T^n$-invariant complete shrinking Kähler-Ricci soliton, and uniqueness holds for solitons with bounded Ricci curvature and the soliton vector field in the Lie algebra $\mathfrak{t}$ of $T^n$ (Cifarelli, 2020).
• The complete classification of two-dimensional shrinking Kähler-Ricci solitons with bounded scalar curvature includes only compact Fano solitons, the flat Gaussian on $\mathbb{C}^2$, the Feldman–Ilmanen–Knopf soliton, the cylinder $\mathbb{C} \times \mathbb{P}^1$, and a unique soliton on the blowup of $\mathbb{C} \times \mathbb{P}^1$ at one point (Bamler et al., 2022).
• Shrinking Sasaki-Ricci solitons satisfy strong topological and compactness results: they are always connected at infinity, and positivity of sectional and Ricci curvature forces compactness (generalizing results of Perelman, Naber, and Munteanu–Wang for lower-dimensional and Kähler cases) (Chang et al., 19 Aug 2025).

7. Summary of Key Formulas

• Soliton equation: $\mathrm{Ric} + \nabla^2 f = \frac{1}{2}g$
• Gaussian shrinker: $f(x) = \frac{|x|^2}{4}$, $(\mathbb{R}^n,g_{\mathrm{Eucl}})$
• Product splitting (high-dimensional, Bach-flat): $ds^2 = dr^2 + \phi(r)^2\,g_N$, $N$ Einstein
• Bach tensor in terms of D-tensor: $B_{ij} = \frac{1}{n-2}(\nabla_k D_{ikj} + \cdots)$
• D-tensor/Cotton/Weyl relation: $D_{ijk} = C_{ijk} + W_{ijkl}\nabla_l f$

Conclusion

The theory of shrinking gradient Ricci solitons reveals that under natural curvature, conformal, or analytic conditions, the space of solitons is strikingly rigid. In dimension four, this leads to a complete classification under Bach-flatness, positive isotropic curvature, or pinching conditions involving the Weyl tensor: only the standard models (flat, cylinder, sphere, and their quotients or products) are permitted. In higher dimensions, curvature harmonicity results and warped product splittings further restrict the possible geometries. For Kähler and Sasaki cases, similar rigidity and uniqueness phenomena occur, with toric symmetry and asymptotic geometry playing key roles. These results solidify the position of shrinking solitons as both structurally rigid and archetypically simple representatives of singularity models in Ricci flow.