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Harmonic map

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional called the Dirichlet energy. As such, the…

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Overview

Geometry of mappings between manifolds

Dirichlet energy and its variation formulas

Examples of harmonic maps

Harmonic map heat flow

The Bochner formula and rigidity

Problems and applications

Harmonic maps between metric spaces

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Map overview Semantic statistics

Harmonic map

Nodes94
Edges93
Triples51
Avg. degree1.98
Density0.021277
Components1

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Harmonic map

Top relations

related to Singularities and weak solutions · 23
Harmonic map → Chang, Ding, Dirichlet, Eells, Following, For, Gang Tian, In, Kung-Ching Chang, Michael Struwe, Modeled, Moreover, On, Rugang Ye, Sacks, Sampson's, Since, Struwe's, Their, Uhlenbeck
has application · 6
Harmonic map → Dirichlet, Existence, In, Once, One, Riemannian
related to The Bochner formula and rigidity · 6
Harmonic map → Bochner, Eells, Sampson's, Suppose, The, This
related to Well-posedness · 6
Harmonic map → Eells, Let, Riemannian, Sampson, Tft, This
related to Eells and Sampson's theorem · 5
Harmonic map → Eells, In, Sampson's, The, This
related to External links · 4
Harmonic map → Harmonic, Harmonic Morphisms, Maps BibliographyThe Bibliography, MathWorld
is a · 1
Harmonic map → critical point of the Dirichlet energy

Important terminology Word statistics

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Important terminology

harmonic map manifolds smooth maps energy flow riemannian heat dirichlet mapping eells one given curvature existence geometric differential theorem formula

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Harmonic mapis acritical point of the Dirichlet energy0.90text
Harmonic maphas applicationExistence0.60section
Harmonic maphas applicationOnce0.60section
Harmonic maphas applicationOne0.60section
Harmonic maphas applicationIn0.60section
Harmonic maphas applicationDirichlet0.60section
Harmonic maphas applicationRiemannian0.60section
Harmonic maprelated to Eells and Sampson's theoremThe0.60section
Harmonic maprelated to Eells and Sampson's theoremEells0.60section
Harmonic maprelated to Eells and Sampson's theoremSampson's0.60section
Harmonic maprelated to Eells and Sampson's theoremIn0.60section
Harmonic maprelated to Eells and Sampson's theoremThis0.60section

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