The first and second variation formulas of the energy functional for a nondegenerate map between Finsler manifolds is derived. As an application, some nonexistence theorems of nonconstant stable harmonic maps from a F...The first and second variation formulas of the energy functional for a nondegenerate map between Finsler manifolds is derived. As an application, some nonexistence theorems of nonconstant stable harmonic maps from a Finsler manifold to a Riemannian manifold are given.展开更多
In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact...In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact Finsler manifold M to a com-pact Riemannian manifold with non-positive sectional curvature can be deformed into aharmonic map which has minimum energy in its homotopy class.展开更多
IN this note we prove the existence of the global weak solutions of p-harmonic heat flows, in particular 1【p【2 (for p≥2 see ref. [1]) and extend the strong convergence theorem of Evans to p-heat equations. In parti...IN this note we prove the existence of the global weak solutions of p-harmonic heat flows, in particular 1【p【2 (for p≥2 see ref. [1]) and extend the strong convergence theorem of Evans to p-heat equations. In particular p】1 (see Lemma 2), and it is in itself a significance.展开更多
The aim of this work is to prove the partial regularity of the harmonic maps with potential. The main difficulty caused by the potential is how to find the equation satisfied by the scaling function. Under the assumpt...The aim of this work is to prove the partial regularity of the harmonic maps with potential. The main difficulty caused by the potential is how to find the equation satisfied by the scaling function. Under the assumption on the potential we can obtain the equation, however, for a general potential, even if it is smooth, the partial regularity is still open.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 Brouwer degree of every isoparametric gradient map has been given by using moving frame. It has been used to construct the harmonic representations of the elements of homotopy groups of the spheres.
Some new factorizations of unitons into Lie groups are established via singular Darboux transformations. The factorization processes for Grassmannian unitons are also considered. Furthermore, a purely algebraic method...Some new factorizations of unitons into Lie groups are established via singular Darboux transformations. The factorization processes for Grassmannian unitons are also considered. Furthermore, a purely algebraic method for constructing Grassmannian unitons is presented.展开更多
We obtain the H1-compactness for a system of Ginzburg-Landau equations with pinning functions and prove that the vortices of its classical solutions are attracted to the minimum points of the pinning functions. As a c...We obtain the H1-compactness for a system of Ginzburg-Landau equations with pinning functions and prove that the vortices of its classical solutions are attracted to the minimum points of the pinning functions. As a corollary, we construct a self-similar solution in the evolution of harmonic maps.展开更多
In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a m...In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No.10271106).
文摘The first and second variation formulas of the energy functional for a nondegenerate map between Finsler manifolds is derived. As an application, some nonexistence theorems of nonconstant stable harmonic maps from a Finsler manifold to a Riemannian manifold are given.
基金supported by the National Natural Science Foundation of China(Grant No.10171002).
文摘In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact Finsler manifold M to a com-pact Riemannian manifold with non-positive sectional curvature can be deformed into aharmonic map which has minimum energy in its homotopy class.
文摘IN this note we prove the existence of the global weak solutions of p-harmonic heat flows, in particular 1【p【2 (for p≥2 see ref. [1]) and extend the strong convergence theorem of Evans to p-heat equations. In particular p】1 (see Lemma 2), and it is in itself a significance.
基金supported partly by the National Natural Science Foundation of China(Grant Nos.10371021&10471039)the Natural Science Foundation of Zhejiang Province(Grant No.M103087).
文摘The aim of this work is to prove the partial regularity of the harmonic maps with potential. The main difficulty caused by the potential is how to find the equation satisfied by the scaling function. Under the assumption on the potential we can obtain the equation, however, for a general potential, even if it is smooth, the partial regularity is still open.
基金supported by NSFC (Grant Nos.11971358,11571259,11771339)Hubei Provincial Natural Science Foundation of China (No.2021CFB400)+1 种基金Fundamental Research Funds for the Central Universities (No.2042019kf0198)the Youth Talent Training Program of Wuhan University。
文摘In this article,we prove that a quasi-isometric map between rank one symmetric spaces is within bounded distance from an f-harmonic map.
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Project supported by the National Natural Science Foundation of China,the State Education Commission Foundation of China and the Foundation of the Chinese Academy of Sciences.
文摘The Brouwer degree of every isoparametric gradient map has been given by using moving frame. It has been used to construct the harmonic representations of the elements of homotopy groups of the spheres.
基金Project supported by the Chinese National Research Project"Nonlinear Science",the Scientific Foundation of the National Education Commission of China and the Research Foundation of the Education Commission of Shanghai,the National Natural Science Foundat
文摘Some new factorizations of unitons into Lie groups are established via singular Darboux transformations. The factorization processes for Grassmannian unitons are also considered. Furthermore, a purely algebraic method for constructing Grassmannian unitons is presented.
文摘We obtain the H1-compactness for a system of Ginzburg-Landau equations with pinning functions and prove that the vortices of its classical solutions are attracted to the minimum points of the pinning functions. As a corollary, we construct a self-similar solution in the evolution of harmonic maps.
基金Supported by National Natural Science Foundation of China(Grant No.11471100)。
文摘In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.
