A bridge principle for harmonic maps (Q1973268)

Given two minimal surfaces \(S_0\) and \(S_1\) with nonempty boundaries and a thin strip connecting their boundaries, the bridge principle for minimal surfaces says that if the strip is thin enough, then the new boundary should span a minimal surface which is close to \(S_0\) and \(S_1\) joined by this thin strip. In the paper [Invent. Math. 90, 505-549 (1987; Zbl 0637.49020)], \textit{N. Smale} proved the bridge principle for minimal submanifolds and constant mean curvature submanifolds in \(\mathbb{R}^n\) of arbitrary dimension and codimension by the PDE technique. By applying this procedure, the authors of the present paper provide a bridge principle for harmonic maps between smooth compact submanifolds with boundary of a Riemannian manifold.