Geometric analysis

Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory. More recently, it refers largely to the use of

The scope of geometric analysis includes both the use of geometrical methods in the study of partial differential equations (when it is also known as "geometric PDE"), and the application of the theory of partial differential equations to geometry. It incorporates problems involving curves and surfaces, or domains with curved boundaries, but also the study of Riemannian manifolds in arbitrary dimension. The calculus of variations is sometimes regarded as part of geometric analysis, because differential equations arising from variational principles have a strong geometric content. Geometric analysis also includes global analysis, which concerns the study of differential equations on manifolds, and the relationship between differential equations and topology.

The following is a partial list of major topics within geometric analysis:

• Gauge theory
• Harmonic maps
• Kähler–Einstein metrics
• Mean curvature flow
• Minimal submanifolds
• Positive energy theorems
• Pseudoholomorphic curves
• Ricci flow
• Yamabe problem
• Yang–Mills equations

References

Further reading

• ISBN 978-1-571-46198-8
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• ISBN 978-3-642-16285-5
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• ISBN 978-3-540-25907-7
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• Lee, Jeffrey M. (2009). Manifolds and Differential Geometry.
  ISBN 978-0-8218-4815-9
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• ISBN 978-0-8218-2673-7
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• Helgason, Sigurdur (2008). Geometric Analysis on Symmetric Spaces (2nd ed.). American Mathematical Society.
  ISBN 978-0-8218-4530-1
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