The Maximum Locus of the Bloch Norm

Abstract

Abstract For a Bloch function f in the unit ball in ℂ n , we study the maximal locus of the Bloch norm of f; namely, the set Lf where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set Lf consists of a finite union of real analytic sets with dimensions at most 2n − 2. This is not the case for n = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set Lf . In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.