Course Archives Theoretical Statistics and Mathematics Unit
Course: Special Topics - Geometry: Harmonic maps
Time: Currently not offered
Syllabus
Past Exams

Syllabus:

(i) Preliminaries on smooth manifolds: De nition of smooth manifolds, smooth functions, smooth maps, tangent spaces, derivatives of functions and maps. Vector elds, ows, Lie bracket. Di erential forms, exterior derivative, Stoke's theorem. Prism operator for homotopies, interior contraction with vector elds, Lie derivative of forms, Cartan's formula. (ii) Vector bundles and connections: De nition of vector bundles. Dual, direct sum, tensor product of bundles, pull-back bundles. Connections on vector bundles, covariant di erentiation along curves, parallel transport. Induced connections on induced bundles. (iii) Riemannian geometry: Riemannian metrics, lengths of curves. Submanifolds of Euclidean space with induced metric, variation of lengths of curves, geodesics, induced connection. Levi-Civita connection, geodesics, and exponential map on Riemannian manifolds. Minimal geodesics. First and second variations of energy of curves. Bonnet-Myers and Synge theorems. Divergence of vector elds, divergence theorem. Gradient vector eld and Hessian of a function. Laplacian on a Riemannian manifold. (iv) Harmonic maps: Second fundamental form of a smooth map. Totally geodesic maps. Symmetry of the second fundamental form. Exponential map totally geodesic at the origin. Energy density and energy of maps. First variation of energy and tension eld of a smooth map. Tension eld of a map as Laplacian of the map composed with the inverse of the exponential. Harmonic maps. Examples: geodesics, harmonic functions, minimal isometric immersions, totally geodesic maps. Weitzenbock formula for Laplacian of energy density and applications. Harmonic maps and convex functions, characterization in terms of subharmonicity of composition, maximum principle. Harmonic map heat ow. Monotonicity of energy, convexity of energy for nonpositively curved targets. Short-term existence of solutions to the harmonic map heat ow: equivalence of harmonic maps heat ow with a nonlinear parabolic PDE with values in Euclidean space; short-term existence of solutions for the equivalent PDE via the linearized equation and the Inverse Function Theorem for Banach spaces. Long-term existence of solutions and convergence of solutions to harmonic maps. Eels-Sampson theorem on existence and uniqueness of harmonic maps in a given homotopy class for nonpositively curved and negatively curved targets.

Suggested Texts :
(a) doCarmo, "Riemannian geometry", Birkhauser, 1992.
(b) Xin, "Geometry of harmonic maps", Birkhauser, 1996.
(c) Nishikawa, "Variational problems in geometry", AMS, 2002.

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Past Exams
Midterm Semestral Supplementary and Back Paper
 23.pdf

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