Non-Kähler Shrinking Ricci Solitons

Definitions and mathematical statement

A smooth Riemannian metric $g$ on a manifold $M$ is a shrinking Ricci soliton if there exist a smooth vector field $X$ and a constant $\lambda>0$ such that \[ \operatorname{Ric}_g+\frac12\mathcal L_Xg=\lambda g, \] where $\operatorname{Ric}_g$ is the Ricci tensor and $\mathcal L_Xg$ is the Lie derivative of the metric. In the gradient case, $X=\nabla_g f$ for a smooth function $f:M\to{\mathbb{R}}$, and the equation becomes \[ \operatorname{Ric}_g+\nabla_g^2 f=\lambda g. \] The supplied record does not specify whether only gradient solitons are intended.

A metric $g$ is Kähler if there is a complex structure $J$ on $M$ compatible with $g$ and parallel for its Levi--Civita connection: \[ J^2=-\operatorname{Id},\qquad g(Ju,Jv)=g(u,v),\qquad \nabla^gJ=0. \] Under this metric interpretation, non-Kähler means that no such $J$ exists. The source does not specify whether a complex structure is instead part of the initial data.

Existence questions, with unresolved source qualifications.

1. Does there exist a compact manifold $M$ carrying a non-Kähler shrinking Ricci soliton metric $g$?
2. Does there exist a noncompact manifold $M$ carrying a geodesically complete, non-Kähler shrinking Ricci soliton metric $g$ that is not of Gaussian type?

The original wording joins these alternatives with "or"; the cleaned statement presents them as two questions, which are retained here separately.

For reference, the standard Gaussian gradient shrinker is \[ \left({\mathbb{R}}^n,g_{\mathrm{Euc}},f(x)=\frac{\lambda}{2}|x|^2\right),\qquad \lambda>0. \] This example does not define the source's unspecified phrase "of Gaussian type." In particular, the record does not state whether that phrase includes products with Einstein factors, quotients, or other constructions. The second question therefore has no fully specified exclusion class.

Neither question supplies a dimension restriction, a non-Einstein or nontriviality condition, or a symmetry assumption. The displayed soliton equation permits Einstein metrics with $X=0$. Thus an intended exclusion of such solitons cannot be inferred silently. The supplied background mentions degenerate product examples and a "genuinely non-Gaussian" target, but does not turn these qualifications into mathematical hypotheses. These omissions prevent a unique precise formulation of the intended existence problem.

Short English statement

The record asks whether compact non-Kähler shrinking Ricci solitons exist, and whether complete noncompact examples exist outside a class called Gaussian type. The intended dimension, gradient requirement, exclusion of Einstein or product examples, and meaning of Gaussian type remain unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30002685$, title Non-Kähler Shrinking Ricci Solitons, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2014.
• Supplied source citation: Mini-Workshop: Einstein Metrics, Ricci Solitons and Ricci Flow under Symmetry Assumptions (2015), Oberwolfach Reports, pp. 2529--2568, DOI https://doi.org/10.4171/owr/2014/45. The literature-source list labels this same reference Oberwolfach Report 45/2014. The supplied 2015 citation year and 2014 report designation are retained without reconciliation.
• Supplied literature source: Lower Bounds for Scalar Curvatures of Noncompact Shrinkers, https://doi.org/10.1016/j.crma.2011.11.004. The record does not supply authors or a specific passage establishing the intended formulation.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that nontrivial compact and complete noncompact Kähler shrinkers are known, that product constructions give degenerate examples, and that a non-Kähler shrinker of the requested genuinely non-Gaussian type has not been established as a general solution to this problem. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. This is supplied editorial metadata, not independent confirmation of the mathematical formulation or current open status.