Exponentially harmonic maps of complete Riemannian manifolds

2018 ◽  
Vol 161 (1-2) ◽  
pp. 205-212
Author(s):  
Toshiaki Omori
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Complete Riemannian Manifolds

scholarly journals Nonlinear Dirac Equations, Monotonicity Formulas and Liouville Theorems

2019 ◽  
Vol 372 (3) ◽  
pp. 733-767
Author(s):  
Volker Branding
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Maps ◽  
Qualitative Behavior ◽  
Quantum Field ◽  
Liouville Theorems ◽  
Dirac Equations ◽  
Monotonicity Formulas ◽  
Curvature Term ◽  
Nonlinear Dirac Equations ◽  
Complete Riemannian Manifolds

Abstract We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

scholarly journals UNIQUENESS THEOREMS FOR HARMONIC MAPS INTO METRIC SPACES

2002 ◽  
Vol 04 (04) ◽  
pp. 725-750 ◽  
Author(s):  
CHIKAKO MESE
Keyword(s):  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Harmonic Maps ◽  
Uniqueness Theorems ◽  
Singular Spaces ◽  
Domain Space ◽  
Recent Developments

Recent developments extend much of the known theory of classical harmonic maps between smooth Riemannian manifolds to the case when the target is a metric space of curvature bounded from above. In particular, the existence and regularity theorems for harmonic maps into these singular spaces have been successfully generalized. Furthermore, the uniqueness of harmonic maps is known when the domain has a boundary (with a smallness of image condition if the target curvature is bounded from above by a positive number). In this paper, we will address the question of uniqueness when the domain space is without a boundary in two cases: one, when the curvature of the target is strictly negative and two, for a map between surfaces with nonpositive target curvature.

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