A complete Kähler-Ricci flow coming out of a Kähler cone must be the self-similar soliton flow when it shares the cohomology class, a Killing field, and curvature bounds.
Chow et al., The Ricci flow: techniques and applications
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Uniqueness of asymptotically conical K\"ahler-Ricci flow
A complete Kähler-Ricci flow coming out of a Kähler cone must be the self-similar soliton flow when it shares the cohomology class, a Killing field, and curvature bounds.