Stability on Kähler-ricci Flow, I
نویسنده
چکیده
In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial Kähler metric is very closed to gKE (or gKS) if a compact Kähler manifold with c1(M) > 0 admits a Kähler Einstein metric gKE (or a Kähler-Ricci soliton gKS). The result improves Main Theorem in [TZ3] in the sense of stability of Kähler-Ricci flow. 0. Introduction The Ricci flow was first introduced by R. Hamilton in [Ha]. If the underlying manifold M is Kähler with positive first Chern class c1(M) > 0, it is more natural to study the following Kähler-Ricci flow (normalized), ∂g(t, ·) ∂t = −Ric(g(t, ·)) + g(t, ·), g(0, ·) = g, (0.1) where g is an initial Kähler metric with its Kähler form ωg ∈ 2πc1(M) > 0. It can be shown that (0.1) preserves the Kähler class. Moreover, (0.1) has a global solution gt = g(t, ·) for any t > 0 ([Ca]). So, the main interest and difficulty of (0.1) is to study the limiting behavior of gt as t tends to ∞ (cf. [CT1], [CT2], [TZ3], etc.). In this paper, we study a stability problem of Kähler-Ricci flow (0.1), namely, we assume that M admits a Kähler-Einstein metric or a KählerRicci soliton, and then we analysis the behavior of evolved Kähler metrics gt of (0.1). We shall prove Theorem 0.1 (Main Theorem). Let M be a compact Kähler manifold with c1(M) > 0 which admits a Kähler Einstein metric gKE (or a Kähler-Ricci soliton (gKS,X0) with respect some holomorphic vector field X0 on M) with its Kähler form in 2πc1(M). Let ψ be a Kähler potential of an initial metric g of (0.1) and φ = φt be a family of Kähler potentials of evolved metrics 1991 Mathematics Subject Classification. Primary: 53C25; Secondary: 53C55, 58E11.
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