English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

After establishing the uniqueness of the continuation of local Cauchy datafor harmonic maps between two Riemannian manifolds M and N, we prove (i) areflection principle for a smooth minimal submanifold Y of a Riemannianmanifold M that contains a reflective submanifold of M as a hypersurface and(ii) the reflection principle of for a class of smooth harmonic maps betweenRiemannian manifolds that admit reflective submanifolds. A reflection principlefor harmonic functions on M with respect to a reflective hypersurface of Mfollows easily from the above reflection principle. To consider reflectionprinciples for harmonic and holomorphic maps from general Hermitian symmetricspaces M, the notion of recursive real forms is introduced. Hermitian symmetricspaces admitting such recursive real forms include Hermitian manifolds ofconstant holomorphic curvatures, the complex hyper-quadrics and theirrespective non-compact duals. Let M be an irreducible Hermitian symmetric spaceof complex dimension n with a recursive real form B1 and correspondinginvolutive isometry s1, X be a smooth Riemannian manifold with B2 as areflective submanifold and corresponding involutive isometry s2 . Also let h bea harmonic map from M to X such that (i) h(B1) is in B2 and (ii) h maps tangentvectors 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 aHermitian 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 forharmonic functions and holomorphic functions on the complex plane to harmonicmaps and holomorphic maps from a class of Hermitian symmetric spaces admittingrecursive real forms. A reflection principle for meromorphic functions definedon the complex plane and taking on real values on the real line is also proved.

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