The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk.We first establish the coefficient estimates for polyharmonic mappin...The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk.We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions.Then,using these results,we establish five Landau-type theorems for subclasses of polyharmonic mappings F and L(F),where F has bounded length distortion and L is a differential operator.展开更多
In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,...In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,using these results,we present better estimations for the Bloch constants of certain harmonic mappings L(f),where f is a K-quasiregular harmonic or open harmonic.Finally,we establish three versions of BlochLandau type theorem for biharmonic mappings of the form L(f).These results are sharp in some given cases and improve the related results of earlier authors.展开更多
In this paper,we study a subclass of close-to-convex harmonic mappings whose analytic parts are starlike mappings.We derive some properties and characteristics for this class,such as the bounds of Toeplitz determinant...In this paper,we study a subclass of close-to-convex harmonic mappings whose analytic parts are starlike mappings.We derive some properties and characteristics for this class,such as the bounds of Toeplitz determinants,bounds of Hankel determinants,Zalcman functional and Bohr's inequality.展开更多
In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give ...In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.展开更多
This paper studies the stability of P-harmonic maps and exponentially harmonic maps from Finsler manifolds to Riemannian manifolds by an extrinsic average variational method in the calculus of variations. It generaliz...This paper studies the stability of P-harmonic maps and exponentially harmonic maps from Finsler manifolds to Riemannian manifolds by an extrinsic average variational method in the calculus of variations. It generalizes Li＇s results in [2] and [3].展开更多
In this paper, we investigate biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain some non-existence results for these maps.
By using the simplified method of factorization given by Valli, and the correspondence between the harmonic map φ∶S 2→U(N) and U(N) uniton bundle ν(φ) with energy corresponding to the bundles’ seco...By using the simplified method of factorization given by Valli, and the correspondence between the harmonic map φ∶S 2→U(N) and U(N) uniton bundle ν(φ) with energy corresponding to the bundles’ second Chern class, which is established by Anand, the energy in a case φ∶S 2→U(N) is investigated in order to estimate the energy of a uniton using the uniton number. It is proved that Uhlenbeck’s factorization is energy decreasing. And a method of estimating the energy of a uniton by the uniton number is given.展开更多
The global existence of the heat flow for harmonic maps from noncompact manifolds is considered. When L^m norm of the gradient of initial data is small, the existence of a global solution is proved.
We discuss a class of complete Kaihler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spe...We discuss a class of complete Kaihler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincare inequality case and establish a vanishing theorem provided that the weighted function p is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincare inequality.展开更多
Harmonic mappings from the hexagasket to the circle are described in terms of boundary values and topological data. Explicit formulas are also given for the energy of the mapping. We have generalized the results in [10].
Let Mn be an embedded closed submanifold ofR^(k+1) or a smooth bounded domain inR_(n),where n≥3.We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time,provi...Let Mn be an embedded closed submanifold ofR^(k+1) or a smooth bounded domain inR_(n),where n≥3.We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time,provided that the initial map u0 is in a suitable nontrivial homotopy class with energy small enough.展开更多
The Bloch constants for quasiregular harmonic mappings and open planar harmonic mappings are considered. Better estimates are obtained. The results, presented in this paper, improve the one made by Chen et al. and Gri...The Bloch constants for quasiregular harmonic mappings and open planar harmonic mappings are considered. Better estimates are obtained. The results, presented in this paper, improve the one made by Chen et al. and Grigoryan.展开更多
In this paper, we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, the...In this paper, we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact K¨ahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.展开更多
We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimens...We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m-3.展开更多
Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homoto...Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.展开更多
For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C...For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.展开更多
The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex do...The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).展开更多
By means of the theory of harmonic maps into the unitary group U(N), the authors study harmonic maps into the symplectic group Sp(N). The symplectic uniton and symplectic ex--tended uniton are introduced. The method o...By means of the theory of harmonic maps into the unitary group U(N), the authors study harmonic maps into the symplectic group Sp(N). The symplectic uniton and symplectic ex--tended uniton are introduced. The method of the symplectic Backlund transformation and the Darboux transformation is used to construct new symplectic unitons from a known one.展开更多
The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic ma...The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = （-1, 1）, AU（z）/cosU（z）π/2≤4/π 1/1-｜z｜^2 holds for z E D, where Au（z） is the maximum dilation of U at z. The inequality is sharp for any z E D and any value of U（z）, and the equality occurs for some point in D if and only if U（z） = 4Re {arctan ~a（z）}, z E D, with a M＆bius transformation φa of D onto itself.展开更多
基金supported by the Natural Science Foundation of Guangdong Province(2021A1515010058)supported by the Youth Innovation Foundation of Shenzhen Polytechnic University(6024310023K)。
文摘The main purpose of this paper is to investigate the univalence of normalized polyharmonic mappings with bounded length distortions in the unit disk.We first establish the coefficient estimates for polyharmonic mappings with bounded length distortions.Then,using these results,we establish five Landau-type theorems for subclasses of polyharmonic mappings F and L(F),where F has bounded length distortion and L is a differential operator.
基金supported by the Natural Science Foundation of Guangdong Province(2021A1515010058)。
文摘In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,using these results,we present better estimations for the Bloch constants of certain harmonic mappings L(f),where f is a K-quasiregular harmonic or open harmonic.Finally,we establish three versions of BlochLandau type theorem for biharmonic mappings of the form L(f).These results are sharp in some given cases and improve the related results of earlier authors.
基金Supported by the Natural Science Foundation of Hunan Province(Grant No.2022JJ30185)the Postgraduate Research and Practice Innovation Program of Jiangsu Province(Grant No.KYCX230410)+2 种基金the China Scholarship Council(Grant No.202306840137)the National Natural Science Foundation of China(Grant No.62063029)the Science and Technology Support Project of Pingxiang City(Grant No.2020C0102)。
文摘In this paper,we study a subclass of close-to-convex harmonic mappings whose analytic parts are starlike mappings.We derive some properties and characteristics for this class,such as the bounds of Toeplitz determinants,bounds of Hankel determinants,Zalcman functional and Bohr's inequality.
文摘In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.
基金Supported partially by the NNSF of China(10871171)
文摘This paper studies the stability of P-harmonic maps and exponentially harmonic maps from Finsler manifolds to Riemannian manifolds by an extrinsic average variational method in the calculus of variations. It generalizes Li＇s results in [2] and [3].
基金Supported by the Natural Natural Science Foundation of China(11201400)Supported by the Basic and Frontier Technology Research Project of Henan Province(142300410433)Supported by the Project for Youth Teacher of Xinyang Normal University(2014-QN-061)
文摘In this paper, we investigate biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain some non-existence results for these maps.
文摘By using the simplified method of factorization given by Valli, and the correspondence between the harmonic map φ∶S 2→U(N) and U(N) uniton bundle ν(φ) with energy corresponding to the bundles’ second Chern class, which is established by Anand, the energy in a case φ∶S 2→U(N) is investigated in order to estimate the energy of a uniton using the uniton number. It is proved that Uhlenbeck’s factorization is energy decreasing. And a method of estimating the energy of a uniton by the uniton number is given.
基金Supported by the National Natural Science Foundation of China (1057115610671079+1 种基金10701064)the Zijin Project of Zhejiang University
文摘The global existence of the heat flow for harmonic maps from noncompact manifolds is considered. When L^m norm of the gradient of initial data is small, the existence of a global solution is proved.
文摘We discuss a class of complete Kaihler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincare inequality case and establish a vanishing theorem provided that the weighted function p is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincare inequality.
基金Supported by the grant 08KJD110011,NSK2008/B11,NSK2009/B07,NSK2009/C042008 Jiangsu Government Scholarship for Overseas Studies
文摘Harmonic mappings from the hexagasket to the circle are described in terms of boundary values and topological data. Explicit formulas are also given for the energy of the mapping. We have generalized the results in [10].
基金supported partially by NSFC(Grant Nos.12141103 and 12301074)Guangzhou Basic and Applied Basic Research Foundation(Grant No.2024A04J3637)+1 种基金supported partially by NSFC(Grant No.11971400)National Key Research and Development Projects of China(Grant No.2020YFA0712500)。
文摘Let Mn be an embedded closed submanifold ofR^(k+1) or a smooth bounded domain inR_(n),where n≥3.We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time,provided that the initial map u0 is in a suitable nontrivial homotopy class with energy small enough.
基金supported by the Research Foundation for Doctor Programme (Grant No. 20050574002)the National Natural Science Foundation of China (Grant No. 10471048)
文摘The Bloch constants for quasiregular harmonic mappings and open planar harmonic mappings are considered. Better estimates are obtained. The results, presented in this paper, improve the one made by Chen et al. and Grigoryan.
基金supported by China’s Recruitment Program of Global ExpertsNational Natural Science Foundation of China (Grant No. 11688101)
文摘In this paper, we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact K¨ahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.
文摘We prove a compactness theorem for k-indexed stationary harmonic maps, and show a regularity theorem for this kind of maps which says that the singular set of a k-indexed stationary harmonic map is of Hausdorff dimension at most m-3.
基金supported by the Natural Science Foundation of China(No.11361073)supported by the Natural Science Foundation of Guangxi Province of China(No.2011GXNSFA018127)
文摘Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.
基金supported by the National Science Foundations (Nos. 0700517, 1001115)
文摘For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.
基金supported by National Natural Science Foundation of China (Grant No.10671093)
文摘The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).
基金Project supported by the National Natural Science Foundation of China (No.19531050)the Scientific Foundation of the Minnstr
文摘By means of the theory of harmonic maps into the unitary group U(N), the authors study harmonic maps into the symplectic group Sp(N). The symplectic uniton and symplectic ex--tended uniton are introduced. The method of the symplectic Backlund transformation and the Darboux transformation is used to construct new symplectic unitons from a known one.
基金supported by National Natural Science Foundation of China(Grant No.11071083)
文摘The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = （-1, 1）, AU（z）/cosU（z）π/2≤4/π 1/1-｜z｜^2 holds for z E D, where Au（z） is the maximum dilation of U at z. The inequality is sharp for any z E D and any value of U（z）, and the equality occurs for some point in D if and only if U（z） = 4Re {arctan ~a（z）}, z E D, with a M＆bius transformation φa of D onto itself.
