Geometric analysis

Saddle tower minimal surface. Minimal surfaces are among the objects of study in geometric analysis.

Geometric analysis is a mathematical discipline at the interface of differential geometry and differential equations.

Scope

It 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.

References

• Jost, Jürgen (2005). Riemannian geometry and Geometric Analysis (4th ed.). Springer. ISBN 978-3-540-25907-7.
• Helgason, Sigurdur (2000). Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions) (2nd ed.). American Mathematical Society. ISBN 978-0-8218-2673-7.
• Helgason, Sigurdur (2008). Geometric Analysis on Symmetric Spaces (2nd ed.). American Mathematical Society. ISBN 978-0-8218-4530-1.