Longtime existence of the Kähler-Ricci flow on ℂⁿ

Abstract

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on C n without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete U (n)-invariant Kähler metric with non-negative holomorphic bisectional curvature, and that the solution converges as t → ∞ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on C n which are not necessarily U (n)-invariant.

Kähler-Ricci flow with unbounded curvature

“…This gives an affirmative answer to the above question. The result is related to previous works on the on the existence of Kähler-Ricci flows without curvature bound, see [3,4,10,24] for example.…”

Kähler–Ricci flow of cusp singularities on quasi projective varieties

“…It can be proved that there is a unique solution g(t) satisfying (1.7) and (1) on the time interval [0, c 4n K ), though it is not known if our solution is unique such on the whole time interval [0, T [ω 0 ] ). Uniqueness of complete bounded curvature solutions to the real Ricci flow (in particular (1.1)) was proved in [6] and for a more general class of complete solutions to (1.1) in [5].…”

An existence time estimate for Kähler–Ricci flow

“…We conclude that T ≥ a n T h 0 by letting ǫ → 0. We now prove part (2). Let {g i,0 } ∞ i=1 ⊂ G(h 0 ; a, b, Λ) such that g i,0 → g 0 uniformly in compact sets of M. Then by Theorem 2.3 (2) and Theorem 2.4 (2), we have a sequence of solutions g i (t) to (1.1) with initial data g i,0 on M × (0, aT h 0 ) converging to a smooth limit solution g(t) to (1.1) on M × (0, aT h 0 ).…”

“…We also need the following uniqueness theorem on Kähler-Ricci flow [2]: Theorem A.3. Let (M n , h 0 ) be a complete noncompact Kähler manifold with bounded curvature.…”