An Aubin continuity path for shrinking gradient Kähler-Ricci solitons (Q6629507)

In recent years, Ricci solitons have been studied intensively because they are interesting both from the point of view of canonical metrics and of the Ricci flow (cf. [\textit{X.-J. Wang} and \textit{X. Zhu}, Adv. Math. 188, No. 1, 87--103 (2004; Zbl 1086.53067)]). Moreover they represent one direction in which the concept of an Einstein manifold can be generalized. The authors study the construction of shrinking gradient Kähler-Ricci solitons, models for finite-time Type I singularities of the Kähler-Ricci flow. By setting up an Aubin continuity path for a complex Monge-Ampère equation in a particular geometric setting that allows for control on the data of the equation, the authors construct such solitons and show a solution to the equation for the initial value of the path parameter.\N\NIn the present paper, if \(D\) is a toric Kähler-Einstein Fano manifold, the authors show that any toric shrinking gradient Kähler-Ricci soliton on certain toric blowups of \(\mathbb{C}\times D\) satisfies a complex Monge-Ampère equation. Part (i) of the main theorem determines the soliton vector field of any complete toric shrinking gradient Kähler-Ricci soliton. Parts (ii) and (iii) give a reference metric that is isometric to a model shrinking gradient Kähler-Ricci soliton outside a compact set. Part (iv) gives a complex Monge-Ampère equation that any complete toric shrinking gradient Kähler-Ricci soliton must satisfy with control on the asymptotics of the data of the equation. The main content of the main theorem is part (v), where the authors provide a solution of the complex Monge-Ampère equation corresponding to the solution at the initial value of the path parameter in the Aubin continuity path.\N\NThe authors construct a background metric with the desired properties which leads to the proof of part (ii) of the main theorem. By obtaining the normalization of the Hamiltonian, the proof of part (iii)--(iv) of the main theorem is obtained. For the main part (v) of the main theorem, the authors implement the continuity method. First, the authors give a proof of the Poincaré inequality and derive the a priori weighted energy estimate for the complex Monge-Ampère equation with compactly supported data. Then, by applying the Poincaré inequality, the authors show that the drift Laplacian of the background metric is an isomorphism between polynomially weighted function spaces, and they show the openness part of the continuity argument. The a priori estimates provide a proof for the closedness part. Note that by using the Aubin-Tian-Zhu functionals which appear in the study of Fano manifolds (cf. [\textit{S. Bando} and \textit{T. Mabuchi}, Adv. Stud. Pure Math. 10, 11--40 (1987; Zbl 0641.53065)]) and in the study of shrinking gradient Kähler-Ricci solitons on compact Kähler manifolds (cf. [\textit{G. Tian} and \textit{X. Zhu}, Acta Math. 184, No. 2, 271--305 (2000; Zbl 1036.53052)]), the authors derive a priori energy estimates. At the end of the paper, by proving a local bootstrapping phenomenon for the complex Monge-Ampère equation, the authors establish an a priori weighted estimate at infinity using the Bochner formula in an essential way, which leads to the completion of the main theorem.\N\NThe authors' contributions to the study of shrinking gradient Kähler-Ricci solitons on certain toric blowups in this paper are outstanding. They show a new way to implement the continuity method for deriving a solution of the complex Monge-Ampère equation.