Singular harmonic maps between rank one symmetric spaces of noncompact type
The paper discusses harmonic self-maps of rank one symmetric spaces of noncompact type which solve a Dirichlet problem at infinity. There is an example by \textit{P. Li} and \textit{L. F. Tam} [Ann. Math. (2) 137, No. 1, 167-201 (1993; Zbl 0776.58010)] of a continuous family of harmonic self-maps of the Poincaré upper half plane which all have the identity as boundary trace. This nonuniqueness example is generalized in the current paper, resulting in similar families of harmonic self-maps on every rank one symmetric space of noncompact type. Since uniqueness theorems in this area assume a certain differentiability of the solutions, the nonuniqueness examples have to be less regular. In the present paper, the solutions from the constructed family are Hölder continuous (viewed as self-maps of the compactified spaces) near the boundary with every exponent less than \(1/7\), but not with exponent \(1/2\). Some more refined regularity estimates are also given. The existence proof uses reduction of the harmonic map equation to an ordinary differential equation by an equivariant ansatz. It is essential for the proof that a transformation is found which reduces the ode. to some translation invariant equation, for which each solution is a member of the continuous family of translated solutions.
scientific article; zbMATH DE number 1565566