Explore: Differential Geometric Aspects Of Harmonic Maps

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy

canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed

dissipative half-harmonic flows into spheres: small data in critical Sobolev spaces Half-harmonic gradient flow: aspects of a non-local geometric PDE Well-posedness

is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work are also

Sciences. 56 (3): 208–215. Nijenhuis, Albert (1960), "Geometric aspects of formal differential operations on tensor fields" (PDF), Proc. Internat. Congress

Introduction to Differential Geometry" (second edition, Publish or Perish, 1979). 1985 Robert Steinberg for three papers on various aspects of the theory of algebraic

to partial differential equations, an important example of which is harmonic analysis, where one studies harmonic functions: the kernel of the Laplace

are useful in many problems of geometric analysis due to their regularity properties. In two dimensions, certain harmonic coordinates known as isothermal

preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. Symmetries are not restricted to geometrical objects, but include

point D is the harmonic conjugate of C with respect to A and B precisely if the cross-ratio of the quadruple is −1, called the harmonic ratio. The cross-ratio