The Maximum Locus of the Bloch Norm

For a Bloch function f in the unit ball in ℂⁿ, we study the maximal locus of the Bloch norm of f; namely, the set L_f where the Bergman length of the gradient vector field of f attains its maximum. We prove that for n ≥, the set L_f 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 L_f. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.