×

Harmonic mappings. (English) Zbl 1152.58305

Ji, Lizhen (ed.) et al., Handbook of geometric analysis. No. 1. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-130-8/hbk). Advanced Lectures in Mathematics (ALM) 7, 147-194 (2008).
Summary: This contribution presents a systematic and comprehensive survey of the theory of harmonic mappings. Harmonic mappings were originally introduced as solutions of a non-linear system of partial differential equations that are derived from the variational problem of minimizing the so-called energy, i.e., the \(L^2\)-norm of the derivative of a map between Riemannian manifolds. Therefore, we first discuss those analytic aspects. It turns out that deeper insights into the analysis come from a geometric understanding of underlying convexity properties, and we therefore investigate those. Harmonic mappings have found important applications in Kähler and algebraic geometry which we also present, including a discussion of the difficult, but important case of infinite energy. Finally, we discuss the case of harmonic maps from Riemann surfaces where conformal invariance leads to phenomena that are not present in higher dimensions. We also present an important generalization of harmonic maps from surfaces, the so-called Dirac-harmonic maps that couple the harmonic map equation with a non-linear Dirac equation, preserving conformal invariance. This can be seen as a mathematical version of the non-linear supersymmetric sigma model of quantum field theory. In particular, the theory of harmonic maps has important connections to basic variational problems of modern physics.
For the entire collection see [Zbl 1144.53004].

MSC:

58E20 Harmonic maps, etc.
58E30 Variational principles in infinite-dimensional spaces
35J60 Nonlinear elliptic equations
35K55 Nonlinear parabolic equations
58J35 Heat and other parabolic equation methods for PDEs on manifolds
49S05 Variational principles of physics
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
20G15 Linear algebraic groups over arbitrary fields
14D06 Fibrations, degenerations in algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
32Q15 Kähler manifolds
32Q05 Negative curvature complex manifolds
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H81 Relationships between algebraic curves and physics
81T60 Supersymmetric field theories in quantum mechanics
PDFBibTeX XMLCite