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

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Arkiv för Matematik

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.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China

    Qingyue Liu

  2. Department of Mathematics, Renmin University of China, Beijing, 100872, P. R. China

    Yunyan Yang

Authors
  1. Qingyue Liu
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  2. Yunyan Yang
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Correspondence to Yunyan Yang.

Additional information

This work was partly supported by NSFC 10601065.

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Liu, Q., Yang, Y. Rigidity of the harmonic map heat flow from the sphere to compact Kähler manifolds. Ark Mat 48, 121–130 (2010). https://doi.org/10.1007/s11512-009-0094-4

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  • DOI: https://doi.org/10.1007/s11512-009-0094-4

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