An existence theorem for harmonic mappings of Riemannian manifolds

Gradient estimates and Liouville theorems for Dirac-harmonic maps

“…In the presence of positive target curvature, however, we know since [10] that a restriction on the radius of the target ball is needed in order to obtain estimates. The optimal size of such a ball corresponds to an open hemisphere in the case of the standard sphere, as shown in [10].…”

“…In the presence of positive target curvature, however, we know since [10] that a restriction on the radius of the target ball is needed in order to obtain estimates. The optimal size of such a ball corresponds to an open hemisphere in the case of the standard sphere, as shown in [10]. Remarkably, we can achieve the same optimal condition on the radius R < π/2 √ K N as in [7] for Dirac-harmonic maps as in the original work for harmonic maps.…”

Singularities of harmonic maps

“…Energy minimality readily shows that the w image of the whole ball B 3/4 has small diameter and the regularity of w|B 3/4 holds as in [69]. Since u|B 1/2 = w|B 1/2 the conclusion follows from Schoen's estimate (5) applied to w. Energy minimality is crucial in the above proof as in the original proof of [97] and in the blow-up arguments of [53] and [83].…”

“…Hildebrandt, H. Kaul, and K.O. Widman [69] and others later established results on the existence, uniqueness, and regularity of a weakly harmonic map whose image had small diameter or lies in the domain of a strictly convex function.…”

The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

“…Thus interior regularity may break down, as shown by an example due to Hildebrandt, Kaul, and Widman [HKW,§6]; cf. [EF,Example 12.3].…”