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Limits of solutions to the Kähler-Ricci flow. (English) Zbl 0889.58067

Let \(X\) be a complex manifold and \[ {{\partial g_{i\bar{j}}}\over{\partial z}}=-R_{i\bar{j}}\tag{*} \] the Kähler-Ricci flow.
R. Hamilton pointed out that the behavior of the following two types of limit solutions is very important: a complete solution of (*) is a type II (type III) limit solution if it is defined for \(-\infty<t<\infty\) (\(0<t<\infty\)) with uniformly bounded curvature, nonnegative holomorphic bisectional curvature and positive Ricci curvature where the scalar curvature \(R\) (\(tR\)) assumes its maximum in spacetime. In the Riemannian case, R. Hamilton [J. Differ. Geom. 38, 1-11 (1993; Zbl 0792.53041)] proved that any type II limit solution with positive curvature operator is necessarily a gradient Ricci soliton. Later he conjectured that a similar result should hold in the Kähler case and for type III solutions.
The main result in this paper is the affirmative answer to these conjectures. The main results are: Let \(X\) be a simply connected non-compact complex manifold of dimension \(n\). Then any type II (type III) limit solution to (*) is necessarily a gradient Kähler-Ricci soliton (a homothetically expanding gradient Kähler-Ricci soliton).
The proof is based on Harnack’s estimate for the Kähler-Ricci flow obtained by the author [H.-D. Cao, Invent. Math. 109, 247-263; (1992; Zbl 0779.53043)] and the strong maximum principle. Also, the first example of a one-parameter family of expanding gradient Kähler-Ricci solitons is given.

MSC:

37C10 Dynamics induced by flows and semiflows
53C55 Global differential geometry of Hermitian and Kählerian manifolds
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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