Bergman-harmonic maps of balls

• Elisabetta Barletta ^[1] ; Sorin Dragomir ^[1]
  1. [1] Università della Basilicata

     Università della Basilicata

     Potenza, Italia

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 15, Nº Extra 1 (Special volume), 2016, págs. 269-307
• Idioma: inglés
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• Resumen

  • We study Bergman-harmonic maps between balls 8 : Bn ! BN extending of class either C2 orM1 to the boundary of Bn. For every holomorphic (anti-holomorphic) map 8 : Bn ! BN extending smoothly to the boundary and every smooth homotopy H : 8 ' 9 we prove a Lichnerowicz-type (cf.

    [28]) result, i.e., we show that E✏ (9) # E✏ (8) + O(✏−n+1). When 8 is proper, Bergman-harmonic, and C2 up to the boundary, the boundary values map % : S2n−1 ! S2N−1 is shown to satisfy a compatibility system similar to the tangential Cauchy-Riemann equations on S2n−1 (and satisfied by the boundary values of any proper holomorphic map). For every weakly Bergman-harmonic map 8 2 W1(Bn,BN ) admitting Sobolev boundary values % 2 M1(S2n−1,BN ) in the sense of [6], the boundary values % are shown to be a weakly subelliptic harmonic map of (S2n−1, ⌘) into (BN , h), provided that 8−1rh stays bounded at the boundary of Bn and % has vanishing weak normal derivatives.