Weierstrass-type representation for harmonic maps into general symmetric spaces via loop groups

Integrability of generalized pluriharmonic maps

“…In this section we recall some notions and results on harmonic maps, pluriharmonic maps and propose natural generalizations of the notion of a pluriharmonic map. where ∇ is the connection on T * M ⊗ φ * T N induced by the Levi-Civita connections ∇ g and ∇ h of the metrics g and h. A natural generalization of the concept of a harmonic map [1,28] is to consider a map φ : M → N into an affine manifold (N, ∇ N ) and to replace ∇ in equation (2.1) by the connection induced by ∇ N and ∇ g .…”

“…[7,15,29,43]) and affine symmetric spaces (cf. [1,28]). If one passes from Riemannian surfaces to Kähler manifolds, one has to consider pluriharmonic maps, i.e.…”

“…Further we introduce nearly Kähler manifolds and properties and examples of pluriharmonic Date: August 25, 2018. 1 maps from nearly Kähler sources admitting associated families. In the third section we give a general construction for pluriharmonic maps without associated families from nonintegrable almost complex manifolds (M, J).…”

Regularity for harmonic maps into certain pseudo-Riemannian manifolds

“…Therefore, we would like to explore the extent to which the results developed by Rivière [36] and Rivière and Struwe [39] can be generalized to elliptic systems without an L 2 -antisymmetric structure. Geometrically, considering the link between harmonic maps into S 4 1 ⊂ R 5 1 and the conformal Gauss maps of Willmore surfaces in S 3 (see Bryant [6]; see also [21,35,3,4]), and the regularity results for weak Willmore immersions established by Rivière [37], we are strongly encouraged to find a method to study the regularity for weakly harmonic maps into S 4 1 and then extend it to the cases of more general targets. Physically, it is known that harmonic maps play an important role in string theory (see e.g.…”

“…To extend the notion of generalized (weakly) harmonic maps into the pseudospheres S n ν (1 ν n), we observe that a W 1,1 map from a disc into any of these non-compact targets is not a priori in L ∞ and hence the conservation laws (1.27) make no sense for such a map. Therefore, we need to require that the map u belongs to the Sobolev space W 1, 4 3 so that Finally, we study the regularity for an elliptic system of the form (1.1) with Ω ∈ L 2 (B, so(1, 1) ⊗ ∧ 1 R 2 ) in dimension m = 2 and show by constructing an example that weak solutions in W 1,2 to such an elliptic system might be not in L ∞ .…”