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Weakly biharmonic maps from the ball to the sphere

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Abstract

The aim of this paper is to investigate the existence of proper, weakly biharmonic maps within a family of rotationally symmetric maps \(u_a : B^n \rightarrow {\mathbb {S}}^n\), where \(B^n\) and \({\mathbb {S}}^n\) denote the Euclidean n-dimensional unit ball and sphere respectively. We prove that there exists a proper, weakly biharmonic map \(u_a\) of this type if and only if \(n=5\) or \(n=6\). We shall also prove that these critical points are unstable.

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Acknowledgements

Funding was provided by Ministero dell’Istruzione, dell’Università e della Ricerca (Grant No. PRIN 2015).

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Authors and Affiliations

  1. University of Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Av. Victor Le Gorgeu, B.P. 809, 29285, Brest Cedex, France

    A. Fardoun

  2. Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124, Cagliari, Italy

    S. Montaldo & A. Ratto

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  1. A. Fardoun
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  2. S. Montaldo
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  3. A. Ratto
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Correspondence to S. Montaldo.

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Work supported by Fondazione di Sardegna (Project GESTA) and Regione Autonoma della Sardegna (Project KASBA).

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Fardoun, A., Montaldo, S. & Ratto, A. Weakly biharmonic maps from the ball to the sphere. Geom Dedicata 205, 167–175 (2020). https://doi.org/10.1007/s10711-019-00470-0

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