Reflection Principles for Smooth Harmonic Maps between Riemannian Manifolds and a Schwarz Reflection Principle for Harmonic and Holomorphic Maps from a Class of Hermitian Symmetric Spaces

Abstract

After establishing the uniqueness of the continuation of local Cauchy data for harmonic maps between two Riemannian manifolds M and N, we prove (i) a reflection principle for a smooth minimal submanifold Y of a Riemannian manifold M that contains a reflective submanifold of M as a hypersurface and (ii) the reflection principle of for a class of smooth harmonic maps between Riemannian manifolds that admit reflective submanifolds. A reflection principle for harmonic functions on M with respect to a reflective hypersurface of M follows easily from the above reflection principle. To consider reflection principles for harmonic and holomorphic maps from general Hermitian symmetric spaces M, the notion of recursive real forms is introduced. Hermitian symmetric spaces admitting such recursive real forms include Hermitian manifolds of constant holomorphic curvatures, the complex hyper-quadrics and their respective non-compact duals. Let M be an irreducible Hermitian symmetric space of complex dimension n with a recursive real form B1 and corresponding involutive isometry s1, X be a smooth Riemannian manifold with B2 as a reflective submanifold and corresponding involutive isometry s2 . Also let h be a harmonic map from M to X such that (i) h(B1) is in B2 and (ii) h maps tangent vectors normal to B1 to vectors normal to B2 . We will prove that h(s1(p)) = s2(h(p)) all p in M. A special case of the above assertion is that X is a Hermitian symmetric space, B_2 is a real form of X and h is a holomorphic map. These two assertions generalize the classical Schwartz reflection principle for harmonic functions and holomorphic functions on the complex plane to harmonic maps and holomorphic maps from a class of Hermitian symmetric spaces admitting recursive real forms. A reflection principle for meromorphic functions defined on the complex plane and taking on real values on the real line is also proved.