A note on compact Kähler-Ricci flow with positive bisectional curvature

Huai Dong Cao; Meng Zhu

doi:10.4310/MRL.2009.v16.n6.a2

A note on compact Kähler-Ricci flow with positive bisectional curvature

Huai Dong Cao^*
, Meng Zhu

^*Corresponding author for this work

Lehigh University

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We show that for any solution g_ij̄ (t) to the Kähler-Ricci flow with positive bisectional curvature R _iījj̄ (t) > 0 on a compact Kähler manifold M ⁿ, the bisectional curvature has a uniform positive lower bound R_iījj̄ (t) > C > 0. As a consequence, g _ij̄ (t) converges exponentially fast in C^∞to a Kähler-Einstein metric with positive bisectional curvature as t → ∞, provided we assume that the Futaki-invariant of Mⁿ is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove [22] in which they assumed the stronger condition that the Mabuchi K-energy is bounded from below.

Original language	English
Pages (from-to)	935-939
Number of pages	5
Journal	Mathematical Research Letters
Volume	16
Issue number	6
DOIs	https://doi.org/10.4310/MRL.2009.v16.n6.a2
State	Published - Nov 2009
Externally published	Yes

Access to Document

10.4310/MRL.2009.v16.n6.a2

Cite this

@article{b0408e31e7df44e2ac254bc72f265ce0,

title = "A note on compact K{\"a}hler-Ricci flow with positive bisectional curvature",

abstract = "We show that for any solution gi{\=j} (t) to the K{\"a}hler-Ricci flow with positive bisectional curvature R iīj{\=j} (t) > 0 on a compact K{\"a}hler manifold M n, the bisectional curvature has a uniform positive lower bound Riīj{\=j} (t) > C > 0. As a consequence, g i{\=j} (t) converges exponentially fast in C∞to a K{\"a}hler-Einstein metric with positive bisectional curvature as t → ∞, provided we assume that the Futaki-invariant of Mn is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove [22] in which they assumed the stronger condition that the Mabuchi K-energy is bounded from below.",

author = "Cao, \{Huai Dong\} and Meng Zhu",

year = "2009",

month = nov,

doi = "10.4310/MRL.2009.v16.n6.a2",

language = "英语",

volume = "16",

pages = "935--939",

journal = "Mathematical Research Letters",

issn = "1073-2780",

publisher = "International Press, Inc.",

number = "6",

}

TY - JOUR

T1 - A note on compact Kähler-Ricci flow with positive bisectional curvature

AU - Cao, Huai Dong

AU - Zhu, Meng

PY - 2009/11

Y1 - 2009/11

N2 - We show that for any solution gij̄ (t) to the Kähler-Ricci flow with positive bisectional curvature R iījj̄ (t) > 0 on a compact Kähler manifold M n, the bisectional curvature has a uniform positive lower bound Riījj̄ (t) > C > 0. As a consequence, g ij̄ (t) converges exponentially fast in C∞to a Kähler-Einstein metric with positive bisectional curvature as t → ∞, provided we assume that the Futaki-invariant of Mn is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove [22] in which they assumed the stronger condition that the Mabuchi K-energy is bounded from below.

AB - We show that for any solution gij̄ (t) to the Kähler-Ricci flow with positive bisectional curvature R iījj̄ (t) > 0 on a compact Kähler manifold M n, the bisectional curvature has a uniform positive lower bound Riījj̄ (t) > C > 0. As a consequence, g ij̄ (t) converges exponentially fast in C∞to a Kähler-Einstein metric with positive bisectional curvature as t → ∞, provided we assume that the Futaki-invariant of Mn is zero. This improves a result of D. Phong, J. Song, J. Sturm and B. Weinkove [22] in which they assumed the stronger condition that the Mabuchi K-energy is bounded from below.

UR - https://www.scopus.com/pages/publications/73849122129

U2 - 10.4310/MRL.2009.v16.n6.a2

DO - 10.4310/MRL.2009.v16.n6.a2

M3 - 文章

AN - SCOPUS:73849122129

SN - 1073-2780

VL - 16

SP - 935

EP - 939

JO - Mathematical Research Letters

JF - Mathematical Research Letters

IS - 6

ER -

A note on compact Kähler-Ricci flow with positive bisectional curvature

Abstract

Access to Document

Other files and links

Fingerprint

Cite this