On pseudo-harmonic maps in conformal geometry

Abstract

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We discuss existence theorems and obtain an extension of Siu's strong rigidity to Kähler–Weyl geometry. Other applications include topological obstructions to the existence of Kähler–Weyl structures. For example, we show that no co‐compact lattice in SO(1, n), n > 2, can be the fundamental group of a compact Kähler–Weyl manifold of certain type.

A Maximum Principle for Generalizations of Harmonic Maps in Hermitian, Affine, Weyl, and Finsler Geometry

“…It is clear that a Weyl harmonic map is also a V -harmonic map with V = − n−2 2 . In [23] the existence of Weyl harmonic maps into a nonpositively curved target was obtained with the method of [22]. Interesting applications on the rigidity were presented in [23] and [24].…”

“…4. In a similar vein, Weyl harmonic maps from a manifold with a Weyl structure into a Riemannian manifold were proposed in [23]. Interesting applications of Weyl harmonic maps have been obtained in [23,24].…”

“…In a similar vein, Weyl harmonic maps from a manifold with a Weyl structure into a Riemannian manifold were proposed in [23]. Interesting applications of Weyl harmonic maps have been obtained in [23,24]. Likewise, affine harmonic maps mapping from an affine manifold into a Riemannian manifold as a new tool for studying affine structures have been introduced in [20,21].…”

Existence and Liouville theorems for V -harmonic maps from complete manifolds

“…The notion of V-harmonic maps was introduced in [5]. It includes the Hermitian harmonic maps introduced and studied in [17], the Weyl harmonic maps from a Weyl manifold into a Riemannian manifold [18], the affine harmonic maps mapping from an affine manifold into a Riemannian manifold [15], [16], and harmonic maps from a Finsler manifold into a Riemannian manifold [2], [12], [30], [33] and [31], see [5] for explanation of these relations. Another interesting special case is when V is a gradient vector field, i.e., V = ∇f for some function f : M → R, then (1.1) takes the form In this case, (1.2) is of divergence form.…”

Locally Conformally Symplectic Structures on Compact Non-Kähler Complex Surfaces

“…lcK metrics for which the Lee form θ is parallel) or more generally, pluricanonical lcK metrics, introduced and studied by G. Kokarev in [29], for which the covariant derivative Dθ of the Lee form is of type (2, 0) + (0, 2) with respect to J, see [37,Claim 3.3] and [40]. 1 The following observation is taken from [36].…”