Harmonic maps into hyperbolic 3-manifolds

Minsky, Yair N.

doi:10.1090/S0002-9947-1992-1100698-9

Harmonic maps into hyperbolic $3$-manifolds
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Trans. Amer. Math. Soc. 332 (1992), 607-632 Request permission

Abstract:

High-energy degeneration of harmonic maps of Riemann surfaces into a hyperbolic $3$-manifold target is studied, generalizing results of [M1] in which the target is two-dimensional. The Hopf foliation of a high-energy map is mapped to an approximation of its geodesic representative in the target, and the ratio of the squared length of that representative to the extremal length of the foliation in the domain gives an estimate for the energy. The images of harmonic maps obtained when the domain degenerates along a Teichmüller ray are shown to converge generically to pleated surfaces in the geometric topology or to leave every compact set of the target when the limiting foliation is unrealizable.

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