The author investigates ancient solutions of the Kähler-Ricci flow \[ \frac{\partial}{\partial t}g_{i\bar j} (x,t) = - \text{Ric}_{i\bar j} (x,t). \] Ancient solutions were introduced in [R. S. Hamilton, Surv. Differ. Geom. 2, 7–136 (1995; Zbl 0867.53030)]. One of geometric invariants associated with ancient solutions is the asymptotic volume ratio \(V (M,g(t))\) which controls the growth of the geodesic ball of radius \(r\) on \((M,g(t))\) at \(r\to\infty\); under some conditions \(V (M,g(t))\) is independent of \(t\) for a family of metrics \(g(t)\) satisfying the Ricci flow equation. The main result of the paper:
Let \((M^m,g(t))\) be a non-flat ancient solution to the Kähler-Ricci flow. Assume that \((M^m,g(t))\) has bounded nonnegative bisectional curvature. Then \(V(M,g(t))=0\). This statement generalises an earlier theorem by Perelman about volume ratio of ancient solutions to the Ricci flow [G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/ 0211159]. The proof is based on the following result about gradient shrinking solitons of Kähler-Ricci flow:
Theorem. Let \((M^m,g)\) be a non-flat gradient shrinking soliton of the Kähler-Ricci flow. (i) If the bisectional curvature of \(M\) is positive then \(M\) must be compact and isometric-biholomorphic to \(P^m\). (ii) If the bisectional curvature of \(M\) is nonnegative then the universal covering of \(M\) splits as \(N_1\times\cdots\times N_l\times C^k\) isometric-biholomorphically, where \(N_i\) are compact irreducible Hermitian symmetric spaces. In particular, \(V(M,g) =0\). This theorem generalises another recent result by G. Perelman [Ricci flow with surgery on three-manifolds, arXiv:math.DG/ 0303109], cf. also T. Ivey [Proc. Am. Math. Soc. 125, 1203–1208 (1997; Zbl 0873.53026)]. Numerous corollaries and applications to the geometry and function theory of complete Kähler manifolds are discussed.