Author(s): Jürgen Jost
Series: Proceedings of the Centre for Mathematical Analysis 4
Publisher: Australian National University
Year: 1983
Language: English
Pages: 188
1. INTRODUCTION
1.1 A short history of variational principles.
1 ~ The concept of geodesics.
1.3 Definition and some elementary properties of harmonic maps.
1.4 Mathematical problems arising from the concep~c of harmonic maps.
1.5 Some examples of harmonic maps.
L 6 Some applica-tions of harmonic maps.
1. 7 Composition properties of harmonic maps.
2. GEOMETRIC PRELIMINARIES Almost linear functions, approximate
fundamental solutions, and r·epr·esentation formulae. Harmonic
coordinates.
2.1 outline of the chapter.
2. 2 Jacobi field estima-tes.
2.3 Applications to geodesic constructions.
2.4 Conv~~ity of geodesic balls.
2.5 The distance as a function of two variables.
2. 6 Almos-t 1 in ear functions.
2.7 Approximate fundamental solutions and representation formulae.
2. 8 Regularity properties of coordinates. Harmonic coordinates.
3. THE HEAT FLOVJ METHOD Existence, regularity, and uniqueness results
for a nonpositively curved image.
3.1 Approaches to the existence and regularity question.
3.2 Short time existence.
3. 3 Estima-tes for the energy density of the heat flow.
3.4 The stability lemma of Hartman.
3.5 A bound for the time derivative.
3.6 Global existence and convergence to a harmonic map (Theorem
of Eells-Sampson) .
3. 7 Es-timates in the elliptic case.
3.8 The uniqueness -results of Hartman.
3.9 The Dirichlet problem.
3 .10 An open question.
4. REGULARITY OF WEAKLY I-IARt10NIC MAPS. Regularity, existence, and
uniqueness of solutions of the Dirichlet problem, if the image
is contained in a convex ball.
4.1 The concept of weak solutions.
4.2 A lemma of Giaquinta-Giusti-Hildebrandt.
4.3 Choice of a test function.
4.4 An iteration argument. Continuity of weak solutions.
4. 5 Holder continuity of weak solutions.
4.6 Applications to the Bernstein problem.
4. 7 Estimates at. the boundary.
4.8 c1 -estimates.
4.9 Higher estimates.
4 .10 The existence theorem of Hildebrandt-Kaul-·Widman.
4.11 'I'he uniqueness theorem of Jager-Kaul.
5. HARMONIC MAPS BETWEEN SURFACES
5.1 Nonexistence results.
5.2 Some lemmata.
5.3 The existence theorem of I"emaireand Sacks-Uhlenbeck.
5. 4 'I'he Dirichlet problem, if the image is homeomorphic to S 2
Two solutions for nonconstant boundary values .
. 5. 5 Conformal diffeomorphisms of spheres. 'I'he Riemann mapping theorem.
5.6 Existence of harmonic diffeomorphisms, if the image is contained
in a convex ball.
5.7 Existence of harmonic diffeomorphisms between closed surfaces.
5.8 Some remarks.
