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**Biharmonic maps on V-manifolds.**
*(English)*
Zbl 1012.58012

A map \(f:M\to N\) between Riemannian manifolds is biharmonic if it is a critical point of the bi-energy functional (integral of the norm-square of the tension field of \(f\)). A smooth V-manifold is a Hausdorff space with an atlas of locally uniformizing systems (roughly, a local uniformizing system in \(M\) is an open set which is homeomorphic to the quotient of an open subset of Euclidean space by the action of a finite group). This generalization of the notion of a manifold is due to I. Satake [Proc. Natl. Acad. Sci. USA 42, 359-363 (1956; Zbl 0074.18103)].

The authors derive the first and second variations for the bi-energy functional with a V-manifold domain and prove that when the target \(N\) has non-positive curvature then \(f\) is biharmonic if and only if it is harmonic. They also show that this cannot be true for manifolds of positive curvature by constructing a non-harmonic biharmonic map into a round sphere. Finally, they show that the post-composition of a biharmonic map by a totally geodesic map is also biharmonic, a result which is well-known for harmonic maps.

The authors derive the first and second variations for the bi-energy functional with a V-manifold domain and prove that when the target \(N\) has non-positive curvature then \(f\) is biharmonic if and only if it is harmonic. They also show that this cannot be true for manifolds of positive curvature by constructing a non-harmonic biharmonic map into a round sphere. Finally, they show that the post-composition of a biharmonic map by a totally geodesic map is also biharmonic, a result which is well-known for harmonic maps.

Reviewer: Ian McIntosh (York)