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Harmonic Maps and Their Existence on Complete Riemannian Manifolds

Author(s) Dr. Indra Kant Jha
Country India
Abstract Harmonic Maps and Their Existence on Complete Riemannian Manifolds Abstract The theory of harmonic maps, initiated by Eells and Sampson in 1964, studies critical points of the Dirichlet energy functional for maps between Riemannian manifolds. When the domain is compact and the target has non-positive sectional curvature, the harmonic map heat flow converges to a smooth harmonic representative in every homotopy class. The situation becomes substantially more delicate when the domain is a complete non-compact Riemannian manifold. In this setting, existence is no longer guaranteed by compactness and requires additional analytic and geometric hypotheses: lower bounds on Ricci curvature, the existence of a positive Green’s function, finite energy or controlled growth of the tension field, and curvature conditions on the target.
This paper surveys the principal existence theorems for harmonic maps from complete Riemannian manifolds. We recall the classical Eells–Sampson heat-flow method and its extensions by Schoen–Yau, Chen–Li, Li–Tam and others. Particular attention is paid to the role of the Bochner formula, gradient estimates, and the interplay between the spectrum of the Laplace–Beltrami operator and the existence of finite-energy harmonic maps. Liouville-type rigidity results, which assert that finite-energy harmonic maps from manifolds with non-negative Ricci curvature into non-positively curved targets must be constant, are also discussed as complementary non-existence statements. The survey concludes with open problems concerning existence under weaker curvature assumptions and the behaviour of the heat flow at infinity.
Keywords Harmonic maps; Complete Riemannian manifolds; Dirichlet energy; Tension field and Non-positive curvature etc.
Published In Volume 1, Issue 1, January-June 2010
Published On 2010-02-06
DOI https://doi.org/10.71097/IJAIDR.v1.i1.2107
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Crossref DOI prefix of IJAIDR is 10.71097/IJAIDR

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