Let M be a compact oriented surface of type \((p,k)\), and \((N,h)\) a complete Riemannian manifold. If \(\mu\) is a conformal structure on M compatible with its orientation, then we write \((M,\mu)\) for the associated Riemann surface. The energy of a map \(\phi:(M,\mu) \to (N,h)\) from the Riemann surface to the Riemannian manifold is \[ (1.1)\quad E(\phi) = \int_{M}| d\phi |^ 2 dx dy. \] Its smooth extrema are called harmonic maps. Let \(\Gamma =(\Gamma_ i)_{1\leq i\leq k}\) be a set of disjoint oriented Jordan curves in N. 1) (N,h) satisfies the uniformity condition (2.6); 2) \(\Gamma\) satisfies the irreducibility condition (2.10); 3) together they satisfy the coercivity condition (2.11). Theorem: If \(\phi_ 0: (M,\mu_ 0)\to (N,h)\) and \(\phi_ 1: (M,\mu_ 1)\to (N,h)\) are homotopic admissible conformal isolated E-minima, then there is a conformal structure \(\mu\) on M and an admissible conformal harmonic map \(\phi: (M,\mu) \to (N,h)\) homotopic to both, which is not an E-minimum. A special case, in which M is a bordered planar domain and N is Euclidean space \({\mathbb{R}}^ n\), is due to \textit{M. Morse} and \textit{C. B. Tompkins} [Ann. Math., II. Ser. 40, 443-472 (1939; Zbl 0021.03405), Duke Math. J. 8, 350-375 (1941; Zbl 0025.40902)] and \textit{M. Shiffman} [Am. J. Math. 61, 853-882 (1939; Zbl 0023.13704); Ann. Math., II. Ser. 40, 834-854 (1939; Zbl 0023.39802); 43, 197-222 (1942)]. If M is a disc (or annulus) and \(N={\mathbb{R}}^ n\), that special case has been reproved by \textit{M. Struwe} [J. Reine Angew. Math. 349, 1-23 (1984; Zbl 0521.49028); ibid. 368, 1-27 (1986; Zbl 0605.58018)]; our proof is an adaptation of his. In Section 3 we adjust the theory of critical points to include suitably differentiable functions on closed convex subsets of Banach spaces. The key technical property is a version of the compactness condition of Palais-Smale. That insures the validity of the mountain pass Proposition (3.12). And of a form of Morse's inequalities (3.13, 3.14) if all critical points are isolated. We give a proof of the Theorem in Section 4, in case M is a disc; the proof in the general case is given in Section 5.