From Stein manifolds to Oka manifolds: the h-principle in complex analysis

2025/09/25 by Forstneric, Franc · 1 citation
#14R10 #53D35 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Primary 32Q56. Secondary 32H02

paper · doi:10.48550/arxiv.2509.21197

Abstract

This introduction to the homotopy principle in complex analysis and geometry, better known as the Oka theory, is aimed at wide mathematical audience. After a brief historical survey of the h-principle in smooth analysis and geometry, I present the key notions of Oka manifolds and Oka maps, which developed from the Oka-Grauert principle and Gromov's theory of elliptic complex manifolds and elliptic holomorphic submersions. I discuss recent and ongoing developments, open problems, and mention some applications. The paper also includes a brief survey of the recently developed h-principles in the classical theory of minimal surfaces.

Citations

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• Model structures and the Oka Principle
• Noncritical holomorphic functions on Stein manifolds
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• The Oka principle for multivalued sections of ramified mappings
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• Handlebody Construction of Stein Surfaces

Cited by

• Parametric equivariant Oka principle

Related

• The homotopy principle in complex analysis: a survey
• Mapping Cylinders and the Oka Principle
• Eight lectures on Oka manifolds
• Oka's principle for holomorphic submersions with sprays
• Recent developments on Oka manifolds