Calabi flow on toric varieties with bounded Sobolev constant, I

Abstract

Let (X, P ) be a toric variety. In this note, we show that the C 0 -norm of the Calabi flow ϕ(t) on X is uniformly bounded in [0, T ) if the Sobolev constant of ϕ(t) is uniformly bounded in [0, T ). We also show that if (X, P ) is uniform K-stable, then the modified Calabi flow converges exponentially fast to an extremal Kähler metric if the Ricci curvature and the Sobolev constant are uniformly bounded. At last, we discuss an extension of our results to a quasi-proper Kähler manifold.

Calabi flow on projective bundles, I

“…Now we are ready to prove 1.5 which is the following theorem: Theorem 4.3. There exists a constant C > 0 independent of t such that for any f ∈ C ∞ ( P ) and for any t ∈ [0, T ), one has By adapting the results in [21], we obtain a proof of 1.7:…”

“…Proof of 1.7. Once we are able to control the Sobolev constant along the Calabi flow, we can apply the techniques developed in [21] to get the uniform C 0 -norm bounds. Let ϕ CP 2 (t), t ∈ [0, T ) be the Calabi flow on X with uniform Sobolev constants bounds.…”

“…By 3.2, there exists a constant C > 0 independent of t such that P u 2 (t) dµ < C for all t ∈ [0, T ). Then Lemma 3.3 in [21] shows that there exists a constant C independent of t such that min x∈ P u(t, x) > C, for all t ∈ [0, T ). By Proposition 3.4 in [21], it implies that on the complex side, the max of relative Kaehler potentials ϕ CP 2 (t) has a uniform upper bound, i.e.…”

“…Then Lemma 3.3 in [21] shows that there exists a constant C independent of t such that min x∈ P u(t, x) > C, for all t ∈ [0, T ). By Proposition 3.4 in [21], it implies that on the complex side, the max of relative Kaehler potentials ϕ CP 2 (t) has a uniform upper bound, i.e. max where t ∈ [0, T ) and C is some constant independent of t. Also by Corollary 3.7 in [21], there exists some constant C independent of t such that for any t ∈ [0, T ), one has max y∈X ϕ CP 2 (t, y) > C…”

“…The author has initiated another project on the study of the Calabi flow with uniform Sobolev bounds on toric manifolds [21]. By modifying the techniques developed in [21], we obtain the following result: Corollary 1.7 (C 0 -esimate). Following the settings in Theorem 1.5, we have…”