Existence of harmonic maps into CAT(1) spaces

A Liouville-type theorem and Bochner formula for harmonic maps into metric spaces

“…In his thesis, Serbinowski [Ser] established existence, uniqueness, and regularity results for harmonic maps from Riemannian manifolds into CAT(1) spaces under Dirichlet boundary conditions. [BFHMSZ2] proved the existence of harmonic maps from surfaces into CAT(1) spaces for the homotopy problem and therefore generalized the classical Sacks-Uhlenbeck theorem [SU], while [BFHMSZ1] established that harmonic maps from Riemannian polyhedra to CAT(1) spaces are Hölder continuous, and Lipschitz away from the n − 2 skeleton of the domain.…”

Bubble tree convergence for harmonic maps into compact locally CAT(1) spaces

“…These include an ǫ-regularity theorem, a gap theorem, a convergence theorem, and a removable singularity theorem for harmonic maps. Notice that the removable singularity theorem extends [BFHMSZ2,Theorem 3.6], which proved the result for conformal harmonic maps. In Section 3 we prove Theorem 3.6, an isoperimetric inequality for conformal harmonic maps into compact locally CAT(1) spaces with small area and small image.…”

“…In pioneering work, Sacks and Uhlenbeck [SU] determined a priori estimates for critical points to a perturbed energy functional to prove the existence of minimal two-spheres in compact Riemannian manifolds. Recently, Breiner et al [BFHMSZ2] extended this result to the singular setting. Lacking a PDE, they instead used the local convexity of the target space (a locally CAT(1) space) to determine a discrete harmonic map heat flow or harmonic replacement process.…”

“…Renormalizing the domain on the scale of the blow up gives a sequence of finite energy maps with uniform modulus of continuity, which converge in C 0 uniformly on compact sets to the map v : C → X. By proving a removable singularity theorem for conformal harmonic maps, in [BFHMSZ2] they concluded that v is harmonic on S 2 .…”

Harmonic branched coverings and uniformization of CAT($k$) spheres

“…a harmonic map from a sphere) has been a widely influential idea in geometric analysis. The authors of the current article and their collaborators generalized the Sacks-Uhlenbeck theorem in the metric space setting and proved the following [5]:…”