Kähler–Ricci solitons on compact complex manifolds with C1(M) > 0

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Real Monge-Ampere equations and Kahler-Ricci solitons on toric log Fano varieties (2012, result): “For the corresponding results in the presence of a Kähler-Ricci soliton, see [23].”
Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, methodology): “In the case when X is smooth the previous existence result was shown in [20], using a variant of Aubin’s continuity method and the convergence result was shown in [49], using Perelman’s deep estimates [38].”
The Moser-Trudinger inequality on Kähler-Einstein manifolds (2006, background): “In [6] it has been improved to γ = 1/(4n + 5).”
Kähler metrics with constant weighted scalar curvature and weighted K‐stability (2018, unknown): “[12, 16, 82, 83]) fits in to our setting too.”
A Brunn–Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry (2013, methodology): “Of course we need to prove that EZ is well defined, cf [32].”
Li–Yau gradient bounds on compact manifolds under nearly optimal curvature conditions (2015, methodology): “] who derived the Li-Yau bound for weighted manifolds by an ordinary diﬀerential inequality involving the entropy and energy of the backward heat equation. For most recent development, see the papers [CTZ], [Dav], [GM], [LX], [QZZ], [Wan], [WanJ] and the latest [BBG], [ZZ] and references therein. In”