Applications of nonlinear analysis in topology. (English) Zbl 0753.53001

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 261-279 (1991).
[For the entire collection see Zbl 0741.00019.]
This is a condensed survey written by a prominent expert in the field, roughly covering the main achievements from 1978 up to 1990. Leading theorems in this area are the Bott periodicity theorem and the theorems of Hadamard and Myers. The author emphasizes the role of Yau’s proof of the Calabi conjecture and gives a survey of the initial results in the fields of minimal surfaces (the Schoen-Yau theorem, the topological mass theorem, the equivariant loop theorem of Meeks and Yau, the proofs of the Frankel conjecture, the Sacks-Uhlenbeck results on minimal 2-spheres, the sphere theorem of Micallef and Moore), gauge theory and 4-manifolds ( Donaldson’s theory), complex moduli spaces (the Donaldson and Uhlenbeck- Yau theorem on the moduli space of stable bundles on a complex Kähler manifold) and the Poincaré conjecture. The analytical methods used are discussed under the headlines continuity method, borderline dimension, gauge theory and heat equation methods. The paper ends with a paragraph on open problems.


53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57R60 Homotopy spheres, Poincaré conjecture


Zbl 0741.00019