Geometric analysis is amathematical discipline where tools fromdifferential equations, especiallyelliptic partial differential equations (PDEs), are used to establish new results indifferential geometry anddifferential topology. The use oflinear elliptic PDEs dates at least as far back asHodge theory. More recently, it refers largelyto the use ofnonlinear partial differential equations to study geometric and topological properties of spaces, such assubmanifolds ofEuclidean space,Riemannian manifolds, andsymplectic manifolds. This approach dates back to the work byTibor Radó andJesse Douglas onminimal surfaces,John Forbes Nash Jr. onisometricembeddings of Riemannian manifolds into Euclidean space, work byLouis Nirenberg on theMinkowski problem and the Weyl problem, and work byAleksandr Danilovich Aleksandrov andAleksei Pogorelov onconvexhypersurfaces. In the 1980s fundamental contributions byKaren Uhlenbeck,^[1]Clifford Taubes,Shing-Tung Yau,Richard Schoen, andRichard Hamilton launched a particularly exciting and productive era of geometric analysis that continues to this day. A celebrated achievement was the solution to thePoincaré conjecture byGrigori Perelman, completing a program initiated and largely carried out by Richard Hamilton.

The scope of geometric analysis includes both the use ofgeometrical methods in the study ofpartial 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 ofRiemannian manifolds in arbitrary dimension. Thecalculus of variations is sometimes regarded as part of geometric analysis, because differential equations arising fromvariational principles have a strong geometric content. Geometric analysis also includesglobal analysis, which concerns the study of differential equations onmanifolds, and the relationship between differential equations andtopology.

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

• Schoen, Richard;Yau, Shing Tung (2010).Lectures on Differential Geometry. International Press of Boston.ISBN 978-1-571-46198-8.
• Andrews, Ben (2010).The Ricci Flow in Riemannian Geometry: A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem (1st ed.). Springer.ISBN 978-3-642-16285-5.
• Jost, Jürgen (2005).Riemannian geometry and Geometric Analysis (4th ed.). Springer.ISBN 978-3-540-25907-7.
• Lee, Jeffrey M. (2009).Manifolds and Differential Geometry.American Mathematical Society.ISBN 978-0-8218-4815-9.
• 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.

• Differential geometry
• Calculus of variations

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