Variational Boundary Theorem for Maps Into Symmetric Spaces of Noncompact Type

We prove a general theorem concerning critical points of parametric elliptic functionals for maps from a Riemannian orLorentzian manifold into a symmetric space G/K of noncompact type. Let X = G/K be endowed with its canonical G-invariantmetric of nonpositive sectional curvature, and let ¯X denote the Satake compactication with boundary ∂Sat = ¯X \ X. If λ: M → Xis a critical point of a parametric elliptic functional F whose image approaches ∂Sat, then: (I) no smooth extension of λ through the boundary exists with values in the interior of X and non-degenerate pullback; (II) the boundary ∂Sat lies at innite geodesic distance from every interior point; (III) the map λ is smooth in the interior; (IV) in the static case, the solution is unique up to the actionof G; (V) in the dynamic case, the eld equations admit a unique maximal globally hyperbolic development.The proof uses three ingredients: the Satake compactication of G/K; the degeneracy of the G-invariant metric at ∂Sat; andthe Schoen Uhlenbeck regularity theory for harmonic maps into targets of nonpositive curvature. The hyperbolicity theorem uses the wave map theory of Shatah Struwe and Tao, combined with the quasilinear framework of Hughes Kato Marsden.As an application, we specialize to G/K = Sp(2N,R)/ U(N) with the Fisher Bures metric and show that maps arising in quantuminformation geometry have domains that terminate at natural boundaries dictated by the Satake compactication.