We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the (n−2)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points x ∈ X^{(k)} with k ≤ n−2, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of x ∈ X.

Let φ∈C^0∩W_{1,2}(Σ,X) where Σ is a compact Riemann surface, X is a compact locally CAT(1) space, and W_{1,2}(Σ,X) is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map u: Σ → X homotopic to φ or there exists a nontrivial conformal harmonic map v: S^2 → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT(κ) spaces for κ∈{0,1}. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT(κ) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an Lp Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT(κ) spaces. Another consequence we derive from the target variation formula is the Eells–Sampson Bochner formula for CAT(1) targets.

We obtain a residue formula for an obstruction to the existence of coupled Kähler–Einstein metrics described by Futaki–Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with reductive automorphism, does not admit a Kähler–Einstein metric but still admits coupled Kähler–Einstein metrics.

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström (2018) and coupled Kähler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).