scholarly journals Partial regularity of stable p-harmonic maps into spheres

2007 ◽  
Vol 76 (2) ◽  
pp. 297-305
Author(s):  
Min-Chun Hong
Keyword(s):  
Partial Regularity ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Weakly Stable

In this paper we prove partial regularity for a weakly stable p-harmonic map from Ω into Sk when k > 2p – 1.

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 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).

scholarly journals Partial regularity of $p(x)$-harmonic maps

2012 ◽  
Vol 365 (6) ◽  
pp. 3329-3353 ◽  
Author(s):  
Maria Alessandra Ragusa ◽  
Atsushi Tachikawa ◽  
Hiroshi Takabayashi
Keyword(s):  
Partial Regularity ◽  
Harmonic Maps

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 and wild Teichmüller spaces

2019 ◽  
pp. 1-45
Author(s):  
Subhojoy Gupta
Keyword(s):  
Riemann Surface ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Teichmüller Space ◽  
Closed Surface ◽  
Hyperbolic Surface ◽  
Teichmuller Space ◽  
Quadratic Differentials ◽  
Hopf Differential ◽  
Principal Parts

We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has noncompact ends with boundary cusps. This extends Wolf’s parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf differential which determine the geometry of the map near the punctures.

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