scholarly journals $$\alpha $$-Dirac-harmonic maps from closed surfaces

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Jürgen Jost ◽  
Jingyong Zhu
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Nonpositive Curvature ◽  
Existence Problem ◽  
Perturbation Function ◽  
Regularity Theorem ◽  
Closed Surfaces ◽  
Palais Smale Condition ◽  
Target Manifold

Abstract$$\alpha $$ α -Dirac-harmonic maps are variations of Dirac-harmonic maps, analogous to $$\alpha $$ α -harmonic maps that were introduced by Sacks–Uhlenbeck to attack the existence problem for harmonic maps from closed surfaces. For $$\alpha >1$$ α > 1 , the latter are known to satisfy a Palais–Smale condition, and so, the technique of Sacks–Uhlenbeck consists in constructing $$\alpha $$ α -harmonic maps for $$\alpha >1$$ α > 1 and then letting $$\alpha \rightarrow 1$$ α → 1 . The extension of this scheme to Dirac-harmonic maps meets with several difficulties, and in this paper, we start attacking those. We first prove the existence of nontrivial perturbed $$\alpha $$ α -Dirac-harmonic maps when the target manifold has nonpositive curvature. The regularity theorem then shows that they are actually smooth if the perturbation function is smooth. By $$\varepsilon $$ ε -regularity and suitable perturbations, we can then show that such a sequence of perturbed $$\alpha $$ α -Dirac-harmonic maps converges to a smooth coupled $$\alpha $$ α -Dirac-harmonic map.

scholarly journals Harmonic Maps on Kenmotsu Manifolds

2013 ◽  
Vol 21 (3) ◽  
pp. 197-208
Author(s):  
Najma Abdul Rehman
Keyword(s):  
Spectral Theory ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Harmonic Morphisms ◽  
Kenmotsu Manifold ◽  
Kenmotsu Manifolds ◽  
Target Manifold

AbstractWe study in this paper harmonic maps and harmonic morphisms on Kenmotsu manifolds. We also give some results on the spectral theory of a harmonic map for which the target manifold is a Kenmotsu manifold.

scholarly journals Existence of (Dirac-)harmonic Maps from Degenerating (Spin) Surfaces

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Jingyong Zhu
Keyword(s):  
Energy Loss ◽  
Existence Result ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Hyperbolic Surface ◽  
Necessary Condition ◽  
Energy Identity ◽  
Closed Surfaces ◽  
Uniformly Bounded ◽  
Suitable Sequence

AbstractWe study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $$\alpha $$ α -(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial.

Regularity of Dirac-harmonic maps with $$\lambda $$-curvature term in higher dimensions

2019 ◽  
Vol 58 (6) ◽  
Author(s):  
Jürgen Jost ◽  
Lei Liu ◽  
Miaomiao Zhu
Keyword(s):  
Partial Regularity ◽  
Compact Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Higher Dimensions ◽  
Singular Set ◽  
Open Domain ◽  
Dimensional Hausdorff Measure ◽  
Curvature Term ◽  
Target Manifold

Abstract In this paper, we will study the partial regularity for stationary Dirac-harmonic maps with $$\lambda $$λ-curvature term. For a weakly stationary Dirac-harmonic map with $$\lambda $$λ-curvature term $$(\phi ,\psi )$$(ϕ,ψ) from a smooth bounded open domain $$\Omega \subset {\mathbb {R}}^m$$Ω⊂Rm with $$m\ge 2$$m≥2 to a compact Riemannian manifold N, if $$\psi \in W^{1,p}(\Omega )$$ψ∈W1,p(Ω) for some $$p>\frac{2m}{3}$$p>2m3, we prove that $$(\phi , \psi )$$(ϕ,ψ) is smooth outside a closed singular set whose $$(m-2)$$(m-2)-dimensional Hausdorff measure is zero. Furthermore, if the target manifold N does not admit any harmonic sphere $$S^l$$Sl, $$l=2,\ldots , m-1$$l=2,…,m-1, then $$(\phi ,\psi )$$(ϕ,ψ) is smooth.

scholarly journals Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
James Kohout ◽  
Melanie Rupflin ◽  
Peter M. Topping
Keyword(s):  
Constant Curvature ◽  
Harmonic Map ◽  
Gradient Flow ◽  
Minimal Immersion ◽  
Curvature Surface ◽  
Previous Theory ◽  
Target Manifold ◽  
Harmonic Map Flow

AbstractThe harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as {t\to\infty}.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

BOCHNER-TYPE FORMULAS FOR TRANSVERSALLY HARMONIC MAPS

2012 ◽  
Vol 23 (03) ◽  
pp. 1250003 ◽  
Author(s):  
QUN CHEN ◽  
WUBIN ZHOU
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Transverse Curvature ◽  
Sasaki Manifold ◽  
Sasaki Manifolds

The main purpose of this paper is to study the properties of transversally harmonic maps by using Bochner-type formulas. As an application, we obtain the following theorem between compact Sasaki manifolds: Let f be a transversally harmonic map from compact Sasaki manifold M to compact Sasaki manifold M′, and M′ has a strongly negative transverse curvature. If the rank of dTf is at least three at some points of M, then f is contact holomorphic (or contact anti-holomorphic).

A partial regularity theorem for harmonic maps at a free boundary

1989 ◽  
Vol 2 (4) ◽  
pp. 299-343 ◽  
Author(s):  
Frank Duzaar ◽  
Klaus Steffen
Keyword(s):  
Free Boundary ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Regularity Theorem

scholarly journals On VT-harmonic maps

2019 ◽  
Vol 57 (1) ◽  
pp. 71-94 ◽  
Author(s):  
Qun Chen ◽  
Jürgen Jost ◽  
Hongbing Qiu
Keyword(s):  
Maximum Principle ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Existence Results ◽  
Gradient Estimates ◽  
Hermitian Manifolds ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Levi Civita Connection ◽  
Complete Manifolds

Abstract VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jäger–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.

Some results on harmonic and bi-harmonic maps

2017 ◽  
Vol 14 (07) ◽  
pp. 1750098 ◽  
Author(s):  
Ahmed Mohammed Cherif
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Convex Function ◽  
Open Problem ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Complete Riemannian Manifold ◽  
Strongly Convex Function ◽  
Strongly Convex ◽  
Homothetic Vector

In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary [Formula: see text] to Riemannian manifold [Formula: see text] is necessarily constant with [Formula: see text] admitting a strongly convex function [Formula: see text] such that [Formula: see text] is a Jacobi-type vector field (or [Formula: see text] admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.

scholarly journals Harmonic maps with torsion

2020 ◽  
Author(s):  
Volker Branding
Keyword(s):  
Riemannian Manifolds ◽  
Critical Points ◽  
Mathematical Analysis ◽  
Harmonic Maps ◽  
Natural Extension ◽  
Energy Functional ◽  
Target Manifold

AbstractIn this article we introduce a natural extension of the well-studied equation for harmonic maps between Riemannian manifolds by assuming that the target manifold is equipped with a connection that is metric but has non-vanishing torsion. Such connections have already been classified in the work of Cartan (1924). The maps under consideration do not arise as critical points of an energy functional leading to interesting mathematical challenges. We will perform a first mathematical analysis of these maps which we will call harmonic maps with torsion.

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