The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to lie groups and symmetric spaces, focusing on spectral curves for harmonic 2tori. Sym, soliton surfaces and their applications soliton geometry from spectral problems, in. Harmonic maps from ssuperscript 2 to cpsuperscript n. Integrability and vesture for harmonic maps into symmetric. We also introduce harmonic maps on generalized riemannian spaces into hopf algebras and make contact with integrable models in two dimensions. There are examples of nonlinear system systems that are completely. Harmonic maps from or oneconnected domain into and um are treated. Harmonic maps into symmetric spaces and integrable systems 0. The norwegian academy of science and letters has decided to. The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to lie groups and symmetric. Symmetry breaking for the dirichlet problem for harmonic maps from the disc into the 2sphere pierre, morgan, advances in differential equations, 2005. Riemann surfaces and pluriharmonic maps of complex manifolds into compact symmetric spaces from the viewpoint of integrable system the ory, and we shall. Symmetry free fulltext harmonic maps and biharmonic.
Harmonic maps, conservation laws and moving frames. Harmonic maps, loop groups, and integrable systems by martin a. Integrable systems, harmonic maps and the classical theory of surfaces 1994. Darboux transforms of a harmonic inverse mean curvature.
In these terms, is a harmonic map if the rubber, when released but still constrained to stay everywhere in contact with the marble, already finds itself in a position of equilibrium and therefore does not snap into a different shape. Biharmonic maps into compact lie groups and integrable systems urakawa, hajime, hokkaido mathematical journal, 2014. After reading several books and articles about integrable systems, and after several years of work in the field, i consider particularly meaningful the following quotation from frederic heleins book constant mean curvature surfaces, harmonic maps and integrable systems, lectures in mathematics, eth zurich, birkhauser basel 2001. Moreover, the behavior of simple and multisolutions, even in. This map is a diifeomorphism because it is a bijection and. The most unwieldy aspect of the spectral curve description is the periodicity conditions and the second aim is to present four different forms for these. Harmonic maps into symmetric spaces and integrable systems. Developments of harmonic maps, wave maps and yangmills. This is an accessible introduction to some of the fundamental connections between differential geometry, lie groups, and integrable hamiltonian systems. Bringing together experts in the field of harmonic maps and integrable systems to give a coherent account of this subject, this book starts with introductory articles.
Harmonic maps into lie groups, integrable systems and. After formulating the notion of integrability for axially symmetric harmonic maps from. Integrators are discrete dynamical systems that approximate continuous ones. This book provides a thorough introduction to the theory of classical integrable systems, discussing the various approaches to the subject and explaining their interrelations. We explain how results from symplectic and poisson geometry can be used to obtain insight into the dynamics of liouville integrable models. Harmonic maps, loop groups, and integrable systems martin a. Wave maps are harmonic maps on minkowski spaces and have been studied since the 1990s. Terng loop groups and integrable systems, preliminary notes also showed that in some sense, the harmonic map equation arises as a. Harmonic maps from c to a symmetric space continued. Notes on finite dimensional integrable hamiltonian systems. Harmonic maps from t2 to s2 171 26 epilogue 178 references 187 index 193.
Guest and yoshihiro ohnita introduction from the theory of integrable systems it is known that harmonic maps from a riemann surface to a lie group may be studied by in. Wood university of leeds, uk dedicated to the memory of james eells. Minimal surfaces and bubbling analysis an important tool in global analysis, preceding. We focus on one of the most widely studied integrable system, the toda model, and. Onedimensional and twodimensional integrable systems 23 from 2 lax equations to 1 zerocurvature equation 158 24 harmonic maps of finite type 162 25 application. Sterling, title integrable systems, harmonic maps and the classical theory of surfaces, booktitle aspects math. We round of the article by describing how twistor theory and integrable systems can be used to construct many more harmonic maps. Minimal surfaces and bubbling analysis an important tool in global analysis, preceding the work of uhlenbeck, is the palais smale. Thanks to the development of tools from lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically.
Ashton notes taken by dexter chua michaelmas 2016 these notes are not endorsed by the lecturers, and i have modi ed them often. The book begins by introducing the central ideas of the theory of integrable systems, based on lax representations, loop groups and riemann surfaces. In this article we shall provide an exposition on harmonic maps of riemann surfaces and pluriharmonic maps of complex manifolds into compact symmetric spaces from the viewpoint of integrable. Part of the aspects of mathematics book series asma, volume e 23. Dubrovin, integrable systems and riemann surfaces lecture notes, pdf. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. Harmonic maps and integrable systems 3 explain the traditional constructions which use either holonomy of a family of at connections or polynomial killing elds, which are the solutions to the lax pair mentioned above. Surfaces of constant curvature in euclidean space, harmonic maps from surfaces to symmetric spaces, and analogous structures on higherdimensional manifolds are some of the examples that have broadened the horizons of differential geometry, bringing a rich supply of concrete examples into the theory of integrable systems. Harmonic maps into symmetric spaces and integrable systems, year share. Group actions and deformations for harmonic maps martin a. Integrable systems and harmonic maps into lie groups. Integrable systems and harmonic maps into lie groups fergus odea departmentofmathematics, nationaluniversityofireland, galway, ireland. Constant mean curvature surfaces, harmonic maps and integrable. Uhlenbecks major contributions include foundational results on minimal surfaces and harmonic maps, yangmills theory, and integrable systems.
It then presents a guided tour into the theory of completely integrable systems for harmonic maps, followed by two chapters devoted to recent results on the regularity of weak solutions. Citeseerx integrable systems, harmonic maps and the. Harmoic maps of surfaces and integrable system approach a survey, surikaisekikenkyusho kokyuroku 1720 2010, pp127. Hence, when the domain m is compact, there are nonconstant harmonic maps to the circle if and only if the. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. Onedimensional integrable systems 1 lie groups 2 lie algebras 3 factorizations and homogeneous spaces 4 hamiltons equations and hamiltonian systems 5 lax equations 6 adlerkostantsymes 7 adlerkostantsymes continued 8 concluding remarks on onedimensional lax. Harmonic maps into symmetric spaces and integrable system. Distributions on riemannian manifolds, which are harmonic maps choi, booyong and yim, jinwhan, tohoku mathematical journal, 2003. Vertically harmonic maps and harmonic sections of submersions chapter 5. The harmonic map equation also has a hierarchy of negative flows. Harmonic maps, loop groups, and integrable systems.
Constant mean curvature surfaces, harmonic maps and integrable systems. Integrable systems are interesting particularly when the dimension is high. The prototype of the integrable system is the kortewegde vries kdv equation. In some cases they can be used to obtain new properties.
Harmonic maps are the least expanding maps in orthogonal directions. Using spectral curve methods carberrymcintosh proved the surprising result that. The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to lie groups and symmetric spaces, focusing on spectral. Harmonic maps, loop groups, and integrable systems by martin. Onedimensional and twodimensional integrable systems. Notes on finite dimensional integrable hamiltonian systems preliminary version. The rst considers such systems within the framework of hamiltonian mechanics. Some constructions of harmonic maps, 50 years after eellssampson.
The second approach we discuss applies to systems of ordinary. The higher flows of harmonic maps and an application to. Harmonic maps and integrable systems emma carberry abstract. Constant mean curvature surfaces, harmonic maps and.
Harmonic maps as an integrable system springerlink. Graduate textbook on complex analysis, meromorphic functions, harmonic functions, fuchsian groups, riemann surfaces, algebraic curves, riemannroch theorem, weierstrass points, abel theorem, theta function, bakerakhiezer function, kadomtsevpetviashvili kp hierarchy, conformal mappings to disk. Yangmills fields, the critical points of yangmills functionals of connections whose curvature tensors are harmonic, were explored by a few physicists in the 1950s, and biharmonic maps generalizing harmonic maps were introduced by guoying jiang in 1986. Deformations of classical geometries and integrable systems. Old and new integrable cases, in harmonic maps and integrable systems, vol. The core of our work is in chapters 36 where we present the analytical methods. The norwegian academy of science and letters has decided.
Shadi tahvildarzadeh, integrability and vesture for harmonic maps into symmetric spaces arxiv. Phy411 lecture notes part 1 alice quillen september 14, 2018 contents. Harmonic maps are generalisations of the concept of geodesics. Phy411 lecture notes part 1 university of rochester. Harmonic maps are maps between riemannian or pseudoriemannian manifolds which extremize a natural energy integral. With certain conditions, the 2dimensional harmonic map equations are cast as the toda integrable system. Harmonic maps, loop groups, and integrable systems by. Part ii integrable systems theorems with proof based on lectures by a. If a domain manifold m, g is complete and has nonnegative ricci curvature and the sectional curvature of a target manifold n, h is nonpositive, then every energy finite harmonic map is a constant map cf. May 14, 2014 some constructions of harmonic maps, 50 years after eellssampson lund university, 14 may 2014 john c.
We study two approaches to classical integrable systems. Harmoic maps of surfaces and integrable system approach. Harmonic maps, loop groups, and integrable systems book. A presentation of exotic functional spaces from the theory of harmonic analysis is given and these tools are then used for proving regularity results. We also describe a more general approach, for which the existence of both the family a at connections and the lax pair is super. New problems are also arising in mathematical physics. One of the most striking development of the last decades in the study of minimal surfaces, constant mean surfaces and harmonic maps is the discovery that many classical problems in differential geometry including these examples are actually integrable systems. The rst is to give an expository account of the integrable systems approach to harmonic maps from surfaces to lie groups and symmetric spaces, focusing on spectral curves for harmonic 2tori.
Threedimensional special lagrangian cones with crosssection a 2torus all arise from algebraically completely integrable systems constructions. Harmonic maps to the circle s1 are given by integrating harmonic 1forms with integral periods. Complex analysis, riemann surfaces and integrable systems. Harmonic maps and integrable systems pdf free download. Classical integrable systems and linear flow on tori. Kevin costello, edward witten, masahito yamazaki, gauge theory and integrability i, arxiv. Classical integrable systems and linear flow on tori gus schrader honours thesis, november 2009 supervisor. The supersymmetric case is studied and we derive explicit solutions and calculate backlund transformations. Harmonic maps, conservation laws and moving frames frederic. Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Harmonic maps of surfaces and integrable system approach a survey the progress and view of harmonic map theory. Ferreira lisbon which gives a completely explicit formula for all harmonic maps of finite uniton number from a riemann surface to a unitary group, and so all harmonic maps from the twosphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using.
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