Harmonic maps from Riemann surfaces into complex Finsler manifolds

2015 ◽  
Vol 26 (06) ◽  
pp. 1541010
Author(s):  
Seiki Nishikawa
Keyword(s):  
Riemann Surfaces ◽  
Harmonic Maps ◽  
Positive Curvature ◽  
Harmonic Map ◽  
Riemann Sphere ◽  
Energy Functional ◽  
Finsler Metric ◽  
Absolute Minimum ◽  
Holomorphic Quadratic Differential ◽  
Smooth Map

Given a smooth map from a compact Riemann surface to a complex manifold equipped with a strongly pseudoconvex complex Finsler metric, we define the [Formula: see text]-energy of the map, whose absolute minimum is attained by a holomorphic map. A harmonic map is then defined to be a stationary map of the [Formula: see text]-energy functional. We prove that with each harmonic map is associated a holomorphic quadratic differential on the domain, which vanishes if the map is weakly conformal. Also, under the condition that the metric be weakly Kähler, we determine the second variation of the functional, and prove that any [Formula: see text]-energy minimizing harmonic map from the Riemann sphere to a weakly Kähler Finsler manifold of positive curvature is either holomorphic or anti-holomorphic.

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.

Some conditions ensuring the vanishing of harmonic differential forms with applications to harmonic maps and Yang-Mills theory

1982 ◽  
Vol 91 (3) ◽  
pp. 441-452 ◽  
Author(s):  
H. C. J. Sealey
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
Differential Forms ◽  
Finite Energy ◽  
Energy Functional ◽  
Conformal Transformations ◽  
Second Variation ◽  
Yang Mills

In (5) it is shown that if m ≽ 3 then there is no non-constant harmonic map φ: ℝm → Sn with finite energy. The method of proof makes use of the fact that the energy functional is not invariant under conformal transformations. This fact has also allowed Xin(9), to show that any non-constant-harmonic map φ:Sm → (N, h), m ≽ 3, is not stable in the sense of having non-negative second variation.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

2005 ◽  
Vol 16 (09) ◽  
pp. 1017-1031 ◽  
Author(s):  
QUN HE ◽  
YI-BING SHEN
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Second Variation ◽  
Finsler Manifolds ◽  
Totally Geodesic ◽  
The Stability ◽  
The Second Variation ◽  
Weitzenböck Formula

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

scholarly journals Convergence of Discrete Period Matrices and Discrete Holomorphic Integrals for Ramified Coverings of the Riemann Sphere

2021 ◽  
Vol 24 (3) ◽  
Author(s):  
Alexander I. Bobenko ◽  
Ulrike Bücking
Keyword(s):  
Error Estimate ◽  
Branch Point ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Holomorphic Functions ◽  
Convergence Result ◽  
Edge Length ◽  
Ramified Covering ◽  
Compact Riemann Surfaces ◽  
Ramified Coverings

AbstractWe consider the class of compact Riemann surfaces which are ramified coverings of the Riemann sphere $\hat {\mathbb {C}}$ ℂ ̂ . Based on a triangulation of this covering we define discrete (multivalued) harmonic and holomorphic functions. We prove that the corresponding discrete period matrices converge to their continuous counterparts. In order to achieve an error estimate, which is linear in the maximal edge length of the triangles, we suitably adapt the triangulations in a neighborhood of every branch point. Finally, we also prove a convergence result for discrete holomorphic integrals for our adapted triangulations of the ramified covering.

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 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 Regularity of Weak Solution of Variational Problems Modeling the Cosserat Micropolar Elasticity

2020 ◽  
Author(s):  
Yimei Li ◽  
Changyou Wang
Keyword(s):  
Weak Solution ◽  
Harmonic Maps ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Energy Functional ◽  
Geometrically Nonlinear ◽  
Micropolar Elasticity ◽  
One Dimensional ◽  
The Poisson Equation ◽  
System Coupling

Abstract In this paper, we consider weak solutions of the Euler–Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solution is stationary, then its singular set is discrete for $2<p<3$ and has zero one-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.

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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