arXiv:2112.06601v2 [math.DG] 22 Dec 2021 HOW TO CONSTRUCT HARMONIC MAPS TO THE HYPERBOLIC PLANE G. POLYCHROU, E. PAPAGEORGIOU, A. FOTIADIS, C. DASKALOYANNIS Abstract. We study harmonic maps between Riemann surfaces, when the curvature in the target is a negative constant. In [7], harmonic maps are related to the sinh-Gordon equation and a B¨ acklund transformation is introduced, which connects solutions of the sinh-Gordon and sine-Gordon equation. We develop this ma- chinery in order to construct new harmonic maps to the hyperbolic plane. 1. Introduction and Statement of the Results This article has been motivated by the following open problem: how can we construct a harmonic map explicitly? Our aim is to construct harmonic maps between Riemann surfaces, when the curvature in the target is a negative constant, say -1. For that, we develop some of the machinery introduced in [7], which con- nects the harmonic map problem with elliptic versions of the sinh- Gordon and sine-Gordon equation. Let w = w(x,y ) be a solution of the sinh-Gordon equation (1) Δw = 2 sinh(2w) and let u = u(z, ¯ z ) be a solution of the Beltrami equation (2) ∂ ¯ z u ∂ z u = e −2w , where z = x + iy . Then, u is a harmonic map between Riemann sur- faces, when the curvature of the target is -1 [7, Theorem 1, Corollary 2]. In this case, we say that u = R + iS corresponds to w. From now on, consider the target surface fixed and realized as the upper half hy- perbolic plane. Using a specific coordinate system on the domain (see 2010 Mathematics Subject Classification. 35A30, 58J70, 58J72. Key words and phrases. harmonic maps, Beltrami equations, sinh-Gordon equa- tion, sine-Gordon equation, hyperbolic plane, B¨ acklund transformation. 1