Extremal problems for quadratic differentials (Q1913311)

The primary problem studied by the authors is as follows. Let \(D\) be the unit disc with \(n\) \((n\geq 4)\) specified boundary points \(p_\ell\), \(\ell=1,\dots,n\), (vertices) which divide \(\partial D\) into (open) sides. Let \(\mathcal H\) be a free family of homotopy classes \(\gamma_j\), \(j=1,\dots,k\) of paths joining various sides. As in the reviewer's paper [Ann. Math., II. Ser. 66, 440-453 (1957; Zbl 0082.06301)] positive constants \(a_j\), \(j=1,\dots,k\), determine a polygon problem \(P(a_1,\dots,a_k)\) whose solution is provided by a quadratic differential \(Q(z)dj^2\) regular in \(D\cup\partial D\) apart from possible simple poles at the \(p_j\) and real on \(\partial D\). Its trajectory structure consists of quadrangles \(D_j\) associated with the \(\gamma_j\) such that the level curves of \(D_j\) all have length \(a_j\) in the \(Q\)-metric \(|Q(j)|^{{1\over 2}}|dz|\). Let \(b_j\) be the other dimension of \(D_j\) in the \(Q\)-metric. The authors call this the height of the quadrangle, (since we are in two dimensions which might be a better term). Given fixed positive numbers \(b_j\), \(j=1,\dots,k\), the \(a_j\) can be chosen so that these are the corresponding heights. A simple proof is given by the method in the reviewers paper [Tôhoku Mat. J., II. Ser. 45, No. 2, 249-257 (1993; Zbl 0780.30019)]. Now let \(E\) be a compact subset of \(D\) with finitely many non-degenerate components so that \(D\)-\(E\) is a multiply-connected domain. (It is not clear why the authors feel they must stipulate that the components are simply-connected.) Let \(f\) be a conformal mapping of \(D\)-\(E\) into \(D\) such that \(\partial D\) corresponds to itself. This determines corresponding points \(\widehat p\) and homotopy classes \(\widehat\gamma_j\) (actually these classes are determined by the location of the appropriate endpoints). With the same \(b_j\) this determines a quadratic differential \(\widehat Q(z)dj^2\). The problem is to determine \(f\) so that \(\iint |\widehat Q(z)|dA\) (the norm) is maximal. The authors' principal result is that there exists an extremal mapping whose image for \(D\)-\(E\) is admissible with respect to the associated quadratic differential. They claim that such a mapping is unique. The authors, by doubling across the sides on \(\partial D\) pass to a problem in the Teichmüller space of finite bordered Riemann surfaces with a finite number of punctures. They obtain an existence result by using a compactness argument and a variational lemma for the norm of the quadratic differential. They discuss the question of equality. This question can be treated much more easily and with more precision by using the method of the extremal metric. In the primary problem without some normalization an extremal function \(f\) is trivially non-unique (rotations). However, it can also be non-trivially non-unique so that there must be a difficulty somewhere in the authors' considerations. The authors consider some applications to canonical conformal mappings but do not elucidate the uniqueness question in these examples.