## Block Seminar "Harmonic Maps in Geometry"

Organizers: Christian Bär (Potsdam), Bernhard Hanke (Augsburg)

Harmonic maps form a class of maps between Riemannian manifolds which contains various types of maps of great geometric interest: harmonic functions, geodesics, isometric minimal embeddings and many more. They are critical points of an energy functional and are characterized as solutions of a nonlinear elliptic partial differential equation. We will study existence, uniqueness, stability and regularity questions.
Once existence is established (which is not always the case, however), harmonic maps have important geometric applications. They provide information on the fundamental group, on the isometry group, on certain submanifolds and more.

The block seminar took place at the Hotely Srni in Srni, Czech Republic. The participants arrived on Sunday, June 23 in the evening and departed on Friday, June 28 after lunch.

(0.) Introduction (Christian Bär):
Presentation of the topic and overview
(1.) Energy, tension field, and first variation formula (NN): [2, §3.1–§3.3]
(2.) Harmonic maps and the second variation formula (NN): [2, §3.4–§3.5]
(3.) Examples of harmonic maps (NN): [3, §4.3]
(4.) Instability (NN): [3, §5.2]
(5.) Stability of holomorphic maps (NN): [3, §5.3]
(6.) The heat flow method (NN): [2, §4.1]
(7.) Existence of time-local solutions (NN): [2, §4.2]
(8.) Existence of local time-dependent solutions (NN): [2, §4.2]
(9.) Existence of global time-dependent solutions (NN): [2, §4.3]
(10.) Applications: Theorems of Eells-Sampson and Hartman (NN): [2, §4.4]
(11.) Applications: Theorem of Preissmann and complex submanifolds of Kähler manifolds (NN): [2, §4.5 without Thm. 4.28], see also [1, §9.7]
(12.) Harmonic maps defined on surfaces (NN): [1, §10.1]
(13.) Existence in 2 dimensions (NN): [1, §10.2]
(14.) Regularity in 2 dimensions (NN): [1, §10.3]
(15.) Harmonic maps to the unit sphere (NN): Takahashi’s theorem, doCarmo-Wallach theorem [3, pp. 189–191, 194–202]
(16.) Calabi’s theorem (NN): [3, pp. 203–207]

Literature:

 Jürgen Jost, Riemannian geometry and geometric analysis, 7. edition, Universitext, Springer, Cham, 2017.
 Seiki Nishikawa, Variational problems in geometry, Translations of Mathematical Monographs, vol. 205, American Mathematical Society, Providence, RI, 2002.
 Hajime Urakawa, Calculus of variations and harmonic maps, Translations of Mathematical Monographs, vol. 132, American Mathematical Society, Providence, RI, 1993.

The block seminar was supported by: