A conservation law for harmonic maps

Baird, Paul; Eells, James

doi:10.1007/bfb0096222

Cited by 168 publications

(154 citation statements)

References 20 publications

Supporting

Mentioning

145

Contrasting

Unclassified

Order By: Relevance

“…This is a trivial case of harmonic morphism, and similar results for harmonic morphisms into higher dimensinal target has been obtained( [1]. )…”

Section: Corollary 1 the Energy Functional E(σ) Has A Superlinear Gromentioning

confidence: 66%

See 1 more Smart Citation

A dual monotonicity formula for harmonic mappings

Yamada

2003

Calculus of Variations and Partial Differential Equations

View full text Add to dashboard Cite

The well-known monotonicity formula for harmonic maps says that the scaled energy functional over a ball of radius r is a non-decreasing function of r. The proof uses the fact that the energy functional is critical under any compactly supported variation on the domain of the map. In this article, we will instead use the fact that the energy is critical under variations of the map on the image of the map. By choosing the variational vector field suitably it will be shown that a scaled energy considered as an integral functional over a ball of radius r where r is the distance from a point on the image manifold, is monotonically non-decreasing. The formula takes a stronger form when the image is one dimensional.

show abstract

“…This is a trivial case of harmonic morphism, and similar results for harmonic morphisms into higher dimensinal target has been obtained( [1]. )…”

Section: Corollary 1 the Energy Functional E(σ) Has A Superlinear Gromentioning

confidence: 66%

“…It should be remarked that any map to one dimensional target is trivially a harmonic morphism, and there are results similar in spirit to this statement for harmonic morphism into higher dimensional targets (see Eells-Baird [1] for example).…”

Section: Corollary 1 Let U : M → ω Be a Finite Energy Harmonic Map (Vmentioning

confidence: 89%

A dual monotonicity formula for harmonic mappings

Yamada

2003

Calculus of Variations and Partial Differential Equations

View full text Add to dashboard Cite

show abstract

“…Finally, the calculations in Theorem 7.2 give a special case of a result of Eells and Sampson (1964) (see also Baird and Eells (1980) and Eells and Lemaire (1983), p. 43). …”

Section: Remarksmentioning

confidence: 99%

Riemannian submersions and the regular interval theorem of Morse theory

Fischer

1996

Ann Glob Anal Geom

View full text Add to dashboard Cite

A generalized version of the regular interval theorem of Morse theory is proven using techniques from the theory of Riemannian submersions and conformal deformations. This approach provides an interesting link between Riemamnnian submersions (for real valued functions) and Morse theory.Let f: (M,g) -+ R be a smooth real valued function on a noIl-compact complete connected Riemannian manifold (M,g) such that df is bounded in norm away from zero. By pointwise conformally deforming g to pg, p = I(df l 2 , we show that (M, pg) is a complete Riemannian manifold, arid that f: (M, pg) -R is a surjective Riemannian submersion and a globally trivial fiber bundle over R. In particular, all of the level hypersurfaces of f are diffeomorphic, and M is globally diffeomorphic to the product bundle R x f-1(0) by a diffeomorphism F°: R x f-1 (0) -M that "straightens out" the level hypersurfaces of f.Moreover, we show that (FO)*(pg) is a parameterized Riemannian product manifold on Rx f-' (0), i.e., a product manifold with a metric that varies on the fibers {t} x f-'(0). Also, F°: (R x f-1(0), (F°)*(pg)) --(M, g) is a conformal diffeomnorphism between the Rieiannian manifolds (R x f-l(0), (F°)*(pg)) and (M, g), so that (M, g) is confornially equivalent to a parameterized Riemannian product manifold. The conformIlal diffeomorphisIm F°is an isometry between the Riemannian product manifold (R x f-1 (0), 1 + go) (where go is the metric induced by g on f-1(0)) and (M, g) if and only if ldfll = 1 and Hess f = 0.

show abstract

“…This is due to the fact that they can locally be described by (weakly) conformal harmonic immersions ~b: (N 2, h) -+ (M, g) from a Riemann surface. For the dual situation of submersions we have the following result due to Baird and Eells, see [3]: If q~: (M, g) -+ (N 2, h) is a horizontally conformal submersion (see next section) to a surface, then the following conditions are equivalent:…”

Section: Introductionmentioning

confidence: 99%