Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

Paul Laurain 1  and Longzhi Lin 2
  • 1 Institut de Mathématiques de Jussieu, Université Paris Diderot, Bâtiment Sophie Germain, Case 7012, 75205, Paris, France
  • 2 Mathematics Department, University of California – Santa Cruz, 1156 High Street, Santa Cruz, USA
Paul Laurain
  • Institut de Mathématiques de Jussieu, Université Paris Diderot, Bâtiment Sophie Germain, Case 7012, 75205, Paris, Cedex 13, France
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and Longzhi Lin
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  • Mathematics Department, University of California – Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA
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Abstract

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B14 into the sphere 𝕊n. In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into 𝕊n, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm []. Further, we establish the previously unknown result that the energy convexity along the flow yields uniform convergence of the flow.

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