On the structure of support points in the class \(\Sigma\) (Q1088023)

The authors consider the family \(\Sigma\), continuous (complex-valued) linear functionals on it and elements of \(\Sigma\) for which a maximum of \({\mathcal R}L(f)\) occurs, called support points of L. Such an element maps \(\Delta\) : \(| z| >1\) onto a domain admissible with respect to the quadratic differential \[ Q(w)dw^ 2=L(1/(f-w))dw^ 2. \] The authors study the occurence of zeros of \(Q(w)dw^ 2\) on the complement \(\Gamma\) of f(\(\Delta)\). They conjecture that such zeros must be simple. This is not proven but it is shown that if \(\Gamma\) contains a zero of \(Q(w)dw^ 2\) of order at least two it cannot contain a set of two or more arcs emanating from this zero at equi-angular directions. The method is variational. Some further results are obtained in the case of second order zeros. It is shown that no such restrictive statements can be made about the order of zeros not on \(\Gamma\). Finally the authors discuss Ozawa numbers and a conjecture of Kirwan.