Harmonic Maps from Punctured Riemann Surfaces to the Hyperbolic Plane with Prescribed Scherk Asymptotics

Let $X$ be a genus $g\ge 0$ compact Riemann surface that admits an antiholomorphic involution $\iota$ with non empty fixed-point set, and let $D=\{p_1,\dots ,p_k\}\subset \text{Fix}(\iota )$. At each puncture, we prescribe a Scherk map associated to a given real polynomial quadratic differential with even degree and negative leading coefficient. Here a Scherk map is the harmonic diffeomorphism from $\mathbb{C}$ to the interior of an ideal polygon in ${\mathbb{H}}^2$, whose Hopf differential is the given polynomial. We construct a harmonic map $$h:X\mathrm{\backslash }D\to {\mathbb{H}}^2$$ whose asymptotic behavior at each puncture matches the prescribed Scherk map. In particular, the image of $h$ tends to an ideal polygon near each end. The resulting $h$ covers harmonic maps obtained by taking the ${\mathbb{H}}^2$ factors of horizontal catenoids in ${\mathbb{H}}^2\times \mathbb{R}$.