1 Results When a C60 film was irradiated with electron-beam (EB) with an incident energy of 3 kV, a peanut-shaped C60 polymer with metallic properties was formed[1], as shown in Fig.1. To elucidate the origin of the m...1 Results When a C60 film was irradiated with electron-beam (EB) with an incident energy of 3 kV, a peanut-shaped C60 polymer with metallic properties was formed[1], as shown in Fig.1. To elucidate the origin of the metallic properties of the peanut-shaped polymer, we examined the valence photoelectron spectra of the polymer using in situ high-resolution photoelectron spectroscopy and found that the electronic states of the polymer came across the Fermi level (EF)[2]. Interestingly, the spectral shape i...展开更多
In this paper, we derive evolution equation of the integral of the Gauss curvature on an evolving hypersurface. As an application, we obtain a monotone quantity on the level surface of the potential function on a 3-di...In this paper, we derive evolution equation of the integral of the Gauss curvature on an evolving hypersurface. As an application, we obtain a monotone quantity on the level surface of the potential function on a 3-dimensional steady gradient Ricci soliton with positive sectional curvature, and prove that such a soliton is rotationally symmetric outside of a compact set under a curvature decaying assumption. Along the way we will also apply our evolution equation to some other cases.展开更多
We give a new argument on the classification of solutions of Gauss curvature equation on R2,which was first proved by W.Chen and C.Li[Duke Math.J.,1991,63(3):615-622].Our argument bases on the decomposition properties...We give a new argument on the classification of solutions of Gauss curvature equation on R2,which was first proved by W.Chen and C.Li[Duke Math.J.,1991,63(3):615-622].Our argument bases on the decomposition properties of the Gauss curvature equation on the punctured disk.展开更多
In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 ...In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 → G（2, 5） is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.展开更多
It has been shown, under certain conditions on the Gauss curvature, every totally real surface of the Cayley projective plane with parallel mean curvature vector is either flat or totally geodesic.
In this paper,we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(m),which is the case where the generalized Gauss mapΦis ramified over a famil...In this paper,we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(m),which is the case where the generalized Gauss mapΦis ramified over a family of hypersurfaces{Q_(j)}_(j=1)^(q)in P^(m-1)(C)located in the N-subgeneral position.In addition,we investigate the Gauss curvature estimate for the K-quasiconformal harmonic surfaces immersed in R^(3)whose Gauss maps are ramified over a family of hypersurfaces located in the N-subgeneral position.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
The existence and uniqueness of the generalized solution for a kind of nonparametric curvature flow problem are obtained. This kind of curvature flow problem describes the evolution of graphs with speed depending on t...The existence and uniqueness of the generalized solution for a kind of nonparametric curvature flow problem are obtained. This kind of curvature flow problem describes the evolution of graphs with speed depending on the reciprocal of the Gauss curvature.展开更多
In this paper we completely classify the homogeneous two-spheres,especially,the minimal homogeneous ones in the quaternionic projective space HPn.According to our classification,more minimal constant curved two-sphere...In this paper we completely classify the homogeneous two-spheres,especially,the minimal homogeneous ones in the quaternionic projective space HPn.According to our classification,more minimal constant curved two-spheres in HPnare obtained than what Ohnita conjectured in the paper"Homogeneous harmonic maps into complex projective spaces.Tokyo J Math,1990,13:87–116".展开更多
Let M be a compact Riemann surface, h（x） a positive smooth function on M, and Ф（x） a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ（u）=1/2∫M｜△u｜2eФdVg＋...Let M be a compact Riemann surface, h（x） a positive smooth function on M, and Ф（x） a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ（u）=1/2∫M｜△u｜2eФdVg＋8πc∫MueФ-8πclog∫Mheu＋ФdVg.We give a sufficient condition under which J achieves its minimum for c ≤ infx∈MeФ（X）.展开更多
The catenary shells of revolution are widely used in church constructions due to their unique mechanics'features.To have a better understanding of the deformation and stress of the catenary shells of revolution,we...The catenary shells of revolution are widely used in church constructions due to their unique mechanics'features.To have a better understanding of the deformation and stress of the catenary shells of revolution,we formulate the principal radii for two kinds of catenary shells of revolution and their displacement type governing equations.Numerical simulations are carried out based on both Reissner-Meissner(R-M)mixed formulations and displacement formulations.Our investigations show that both deformation and stress response of elastic catenary shells of revolution are sensitive to its geometric parameter c,and reveal that the mechanics of the catenary shells of revolution has some advantages over the spherical shell for some loadings.Two complete codes in Maple are provided.展开更多
We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gra...We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.展开更多
文摘1 Results When a C60 film was irradiated with electron-beam (EB) with an incident energy of 3 kV, a peanut-shaped C60 polymer with metallic properties was formed[1], as shown in Fig.1. To elucidate the origin of the metallic properties of the peanut-shaped polymer, we examined the valence photoelectron spectra of the polymer using in situ high-resolution photoelectron spectroscopy and found that the electronic states of the polymer came across the Fermi level (EF)[2]. Interestingly, the spectral shape i...
基金Supported by National Natural Science Foundation of China (Grant No. 10926062)Advanced Program for Returned Chinese Overseas Scholars by the Department of Human Resources and Social Security of Zhejiang Province
文摘In this paper, we derive evolution equation of the integral of the Gauss curvature on an evolving hypersurface. As an application, we obtain a monotone quantity on the level surface of the potential function on a 3-dimensional steady gradient Ricci soliton with positive sectional curvature, and prove that such a soliton is rotationally symmetric outside of a compact set under a curvature decaying assumption. Along the way we will also apply our evolution equation to some other cases.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11771232).
文摘We give a new argument on the classification of solutions of Gauss curvature equation on R2,which was first proved by W.Chen and C.Li[Duke Math.J.,1991,63(3):615-622].Our argument bases on the decomposition properties of the Gauss curvature equation on the punctured disk.
基金Supported by the National Natural Science Foundation of China (10531090)Knowledge Innovation Funds of CAS (KJCX3-SYW-S03)
文摘In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 → G（2, 5） is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.
文摘It has been shown, under certain conditions on the Gauss curvature, every totally real surface of the Cayley projective plane with parallel mean curvature vector is either flat or totally geodesic.
基金supported by the NFSC(11971182,12271189)the NFS of Fujian Province of China(2019J01066,2021J01304)。
文摘In this paper,we study the value distribution properties of the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(m),which is the case where the generalized Gauss mapΦis ramified over a family of hypersurfaces{Q_(j)}_(j=1)^(q)in P^(m-1)(C)located in the N-subgeneral position.In addition,we investigate the Gauss curvature estimate for the K-quasiconformal harmonic surfaces immersed in R^(3)whose Gauss maps are ramified over a family of hypersurfaces located in the N-subgeneral position.
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.
基金Supported by the Doctorate Foundation of Hebei Education Commission
文摘The existence and uniqueness of the generalized solution for a kind of nonparametric curvature flow problem are obtained. This kind of curvature flow problem describes the evolution of graphs with speed depending on the reciprocal of the Gauss curvature.
基金supported by National Natural Science Foundation of China(Grant Nos.11471299,11401481 and 11331002)。
文摘In this paper we completely classify the homogeneous two-spheres,especially,the minimal homogeneous ones in the quaternionic projective space HPn.According to our classification,more minimal constant curved two-spheres in HPnare obtained than what Ohnita conjectured in the paper"Homogeneous harmonic maps into complex projective spaces.Tokyo J Math,1990,13:87–116".
文摘Let M be a compact Riemann surface, h（x） a positive smooth function on M, and Ф（x） a smooth function on M which satisfies that ∫MeФdVg = 1. In this paper, we consider the functionalJ（u）=1/2∫M｜△u｜2eФdVg＋8πc∫MueФ-8πclog∫Mheu＋ФdVg.We give a sufficient condition under which J achieves its minimum for c ≤ infx∈MeФ（X）.
基金supported by Xi'an University of Architecture and Technology(Grant No.002/2040221134).
文摘The catenary shells of revolution are widely used in church constructions due to their unique mechanics'features.To have a better understanding of the deformation and stress of the catenary shells of revolution,we formulate the principal radii for two kinds of catenary shells of revolution and their displacement type governing equations.Numerical simulations are carried out based on both Reissner-Meissner(R-M)mixed formulations and displacement formulations.Our investigations show that both deformation and stress response of elastic catenary shells of revolution are sensitive to its geometric parameter c,and reveal that the mechanics of the catenary shells of revolution has some advantages over the spherical shell for some loadings.Two complete codes in Maple are provided.
基金supported in part by US NSF-DMS 1016073,NSFC 11271350 and 91130019Special Research Funds for State Key Laboratories Y22612A33S+1 种基金China 863 project 2010AA012301 and 2012AA01A304China 973 project 2011CB309702.
文摘We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework.Utilizing an adaptive finite element implementation with effective gradient recovery techniques,we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters.Numerical examples and applications to the topological analysis of point clouds are also presented.
