## Arkiv för Matematik

• Ark. Mat.
• Volume 48, Number 1 (2010), 121-130.

### Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds

#### Abstract

A Łojasiewicz-type estimate is a powerful tool in studying the rigidity properties of the harmonic map heat flow. Topping proved such an estimate using the Riesz potential method, and established various uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to  $\mathbb{S}^{2}$ (J. Differential Geom. 45 (1997), 593–610). In this note, using an inequality due to Sobolev, we will derive the same estimate for maps from $\mathbb{S}^{2}$ to a compact Kähler manifold N with nonnegative holomorphic bisectional curvature, and use it to establish the uniformity properties of the harmonic map heat flow from $\mathbb{S}^{2}$ to N, which generalizes Topping’s result.

#### Note

This work was partly supported by NSFC 10601065.

#### Article information

Source
Ark. Mat., Volume 48, Number 1 (2010), 121-130.

Dates
Received: 1 February 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907103

Digital Object Identifier
doi:10.1007/s11512-009-0094-4

Mathematical Reviews number (MathSciNet)
MR2594589

Zentralblatt MATH identifier
1191.53028

Rights
2009 © Institut Mittag-Leffler

#### Citation

Liu, Qingyue; Yang, Yunyan. Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds. Ark. Mat. 48 (2010), no. 1, 121--130. doi:10.1007/s11512-009-0094-4. https://projecteuclid.org/euclid.afm/1485907103

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