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.

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.

BUBBLING LOCATION FOR SEQUENCES OF APPROXIMATE f-HARMONIC MAPS FROM SURFACES

2010 ◽  
Vol 21 (04) ◽  
pp. 475-495 ◽  
Author(s):  
YUXIANG LI ◽  
YOUDE WANG
Keyword(s):  
Riemannian Manifold ◽  
Riemann Surface ◽  
Critical Points ◽  
Homotopy Class ◽  
Blow Up ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Identity ◽  
Minimal Point ◽  
Closed Riemann Surface

Let f be a positive smooth function on a closed Riemann surface (M, g). The f-energy of a map u from M to a Riemannian manifold (N, h) is defined as [Formula: see text] and its L2-gradient is: [Formula: see text] We will study the blow-up properties of some approximate f-harmonic map sequences in this paper. For a sequence uk : M → N with ‖τf(uk)‖L2 < C1 and Ef(uk) < C2, we will show that, if the sequence is not compact, then it must blow-up at some critical points of f or some concentrate points of |τf(uk)|2dVg. For a minimizing α-f-harmonic map sequence in some homotopy class of maps from M into N we show that, if the sequence is not compact, the blow-up points must be the minimal point of f and the energy identity holds true.

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.

CRITICAL POINTS OF YANG–MILLS–HIGGS FUNCTIONAL

2011 ◽  
Vol 13 (03) ◽  
pp. 463-486 ◽  
Author(s):  
CHONG SONG
Keyword(s):  
Perturbation Method ◽  
Critical Points ◽  
Existence Result ◽  
Harmonic Maps ◽  
Fiber Bundles ◽  
Energy Identity ◽  
Holomorphic Maps ◽  
Yang Mills ◽  
The One

We use Sacks–Uhlenbeck's perturbation method to find critical points of the Yang–Mills–Higgs functional on fiber bundles with 2-dimensional base manifolds. Such critical points can be regarded as a generalization of harmonic maps from surfaces, and also a generalization of the so-called twisted holomorphic maps [15]. We prove an existence result analogous to the one for harmonic maps. In particular, we show that the so-called energy identity holds for the Yang–Mills–Higgs functional.

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 A class of factorable topological algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 53-59 ◽  
Author(s):  
E. Ansari-Piri
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Approximate Identity ◽  
Necessary Condition ◽  
Topological Algebras ◽  
Approximate Identities ◽  
Cohen Factorization ◽  
Locally Convex ◽  
Space Property ◽  
Uniformly Bounded

The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

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.

scholarly journals DISTORTION OF SPHERES AND SURFACES IN SPACE

2020 ◽  
Vol 71 (3) ◽  
pp. 981-988
Author(s):  
Sebastian Baader ◽  
Luca Studer ◽  
Roger Züst
Keyword(s):  
Lower Bound ◽  
Unit Disc ◽  
Closed Surfaces ◽  
Positive Genus ◽  
Sharp Lower Bound ◽  
Uniformly Bounded

Abstract It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb{R}^3$. In this note we show that distortion minimizers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $\pi /2$ is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.

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