Some remarks on the class of Riemann surfaces with \((W)\)-property (Q1318947)

The author recalls the \((W)\)-property (in honor to M. Watanabe) for Riemann surfaces. This is related to real square integrable harmonic differentials. If \(P_ W\) denotes the class of Riemann surfaces satisfying the \((W)\)-property, then \(O_{KD}= O_{AD} \cup P_ W\) (a result trivially valid for finite genus surfaces). For each 1-cycle \(c\) on a Riemann surface \(R\), there are attached to it (in a natural way) two harmonic real differentials \(\sigma_{hse} (c)\) and \(\sigma_{ho} (c)\). The main result is that, for a Riemann surface \(R\), we have that \(R\in P_ W\) is equivalent to the property that for each 1-cycle \(c\) on \(R\) one has \(\| \sigma_{hse} (c)\|_ R= \| \sigma_{ho} (c)\|_ R\). In particular, the author gives an equivalent condition for \(R\) to be of type \(O_{AD}\) when \(R\) is a finite genus surface. An explicit example answering a question of Marden is given. In general, a very explicit paper.