The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ ...The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ λj＋ nH）,S = ∑i（ λi＋ nH）~2, where h（ij）= λiδ（ij）. Utilizing Lagrange＇s method, a sharper pointwise estimation of 3（A- 2B） in terms of S and ｜▽h｜~2 is obtained, here ｜▽h｜~2= ∑i,j,k h（ijk）~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.展开更多
In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4（-1）（H5（-1）） with constant mean curvature H satisfying H2 6634 （H2 114785 respectively） must be compact. Speciall...In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4（-1）（H5（-1）） with constant mean curvature H satisfying H2 6634 （H2 114785 respectively） must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4（-1） （resp. H5（-1）） with constant mean curvature H satisfying H2 6643 （resp. H2 114785 ） must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].展开更多
In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces＇ geometric invariants. This extends the previous similar result on compact m...In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces＇ geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.展开更多
Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（...Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.展开更多
Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constan...Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).展开更多
Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where...Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where 5" is the square of the length of the second fundamental form of M,‖·‖<sub>2</sub> denotes the L<sup>2</sup>-norm on M.In this paper,we generalize Hsu’s result to any compact surfaces in S<sup>3</sup> with constant mean curvature.展开更多
By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stabl...By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.展开更多
In this paper, we focus on a constant elasticity of variance （CEV） modeland want to find its optimal strategies for a mean-variance problem under two constrainedcontrols： reinsurance/new business and investment （n...In this paper, we focus on a constant elasticity of variance （CEV） modeland want to find its optimal strategies for a mean-variance problem under two constrainedcontrols： reinsurance/new business and investment （no-shorting）. First, aLagrange multiplier is introduced to simplify the mean-variance problem and thecorresponding Hamilton-Jacobi-Bellman （HJB） equation is established. Via a powertransformation technique and variable change method, the optimal strategies withthe Lagrange multiplier are obtained. Final, based on the Lagrange duality theorem,the optimal strategies and optimal value for the original problem （i.e., the efficientstrategies and efficient frontier） are derived explicitly.展开更多
In this note, we shall prove an interesting result.Theorem. Let 2 be a piece of surface without an umbilical point in 3-dimensional constant curvature space M<sup>3</sup>（C） and possess a constant mean c...In this note, we shall prove an interesting result.Theorem. Let 2 be a piece of surface without an umbilical point in 3-dimensional constant curvature space M<sup>3</sup>（C） and possess a constant mean curvature C<sub>1</sub> （C<sub>1</sub>】0）. ∑ can be isometric to a piece of the surface ∑<sup>*</sup> without an umbilical point, ∑<sup>*</sup> owning a constant mean curvature C<sub>2</sub>（C<sub>2</sub>】0 and C<sub>1</sub>≠C<sub>2</sub>） in M<sup>3</sup>（C）展开更多
In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean φ=0.6180339887, yielding for dark energy to matte...In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean φ=0.6180339887, yielding for dark energy to matter mass fractions .?Assuming the baryonic matter to be only 4.432%, the ratio of matter to baryonic matter would be , and further the ratio of dark matter to baryonic one . If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajdukovic, the remaining mass ratio would yield . Replacing the “Madelung” constant α of Villata’s “lattice universe” by φ, one reaches again 1 + φas the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant and φ−1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the square of the most irrational universal number φmay indicate that the chaotic cosmological processes have reached a quite stable equilibrium. This may be confirmed by another, but similar representation of the mass constituents by the Archimedes’ constant π, giving for respectively for the dark components . However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.展开更多
In this peper the authors give the curvature estimates for complete strongly stable surfaces with constant mean curvature in space forms and extend the related results.
In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piece...In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.展开更多
Using a new set of nucleon coupling constants CZll the properties of a proto neutron star are examined within the framework of the relativistic mean-field theory for the baryon octet system. It is found that the relat...Using a new set of nucleon coupling constants CZll the properties of a proto neutron star are examined within the framework of the relativistic mean-field theory for the baryon octet system. It is found that the relative number density of A,≡ , and ≡0 for CZll are all smaller than those for GL97 and for both CZ11 and GL97, ∑-∑0 and ∑＋ do not appear. It is also found that the pressure and the maximum mass for CZll are all smaller than those for GL97. The maximum mass for CZ11 decreases by approximately 9 percent compared with that for GL97.展开更多
Sommerfeld’s fundamental fine-structure constant α once more gives reason to be amazed. This comment is a Chapter of a publication in preparation dealing mainly with golden ratio signature behind Preston Guynn’s fa...Sommerfeld’s fundamental fine-structure constant α once more gives reason to be amazed. This comment is a Chapter of a publication in preparation dealing mainly with golden ratio signature behind Preston Guynn’s famous matter/space approach. As a result we present a relation of α to the galactic velocity , mediated by the circle constant π, which points to an omnipresent importance of this constant and its intrinsic reciprocity pecularity: α ≈ π<sup>2</sup>|β<sub>g</sub>| respectively . The designation fine-structure constant should be replaced simply by Sommerfeld’s constant. We present golden mean-based approximations for α as well as for electron’s charge and mass and connect the word average value of interaction coupling constant α<sub>s</sub>(m<sub>z</sub>) with |β<sub>g</sub>|.展开更多
文摘The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ λj＋ nH）,S = ∑i（ λi＋ nH）~2, where h（ij）= λiδ（ij）. Utilizing Lagrange＇s method, a sharper pointwise estimation of 3（A- 2B） in terms of S and ｜▽h｜~2 is obtained, here ｜▽h｜~2= ∑i,j,k h（ijk）~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.
基金supported by NSFC (10901067)partially supported by NSFC (10801058) and Hubei Key Laboratory of Mathematical Sciences
文摘In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4（-1）（H5（-1）） with constant mean curvature H satisfying H2 6634 （H2 114785 respectively） must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4（-1） （resp. H5（-1）） with constant mean curvature H satisfying H2 6643 （resp. H2 114785 ） must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].
文摘In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces＇ geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.
基金Supported by the National Natural Science Foundation of China (10771187 10671087)+1 种基金Trans-Century Training Programme Foundation for Talents by the Ministry of Education of ChinaJiangxi Province Natural Science Foundation (2008GZS0060)
文摘Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.
基金supported by National Natural Science Foundation of China(Grant No.11531012)China Postdoctoral Science Foundation(Grant No.BX20180274)Natural Science Foundation of Zhejiang Province(Grant No.LY20A010024)。
文摘Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).
文摘Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where 5" is the square of the length of the second fundamental form of M,‖·‖<sub>2</sub> denotes the L<sup>2</sup>-norm on M.In this paper,we generalize Hsu’s result to any compact surfaces in S<sup>3</sup> with constant mean curvature.
基金The first author is partially supported by the National Natural Science Foundation of China (No.10271106)The second author is partially supported by the 973-Grant of Mathematics in China and the Huo Y.-D. fund.
文摘By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.
基金The NSF(11201111) of ChinaHebei Province Colleges and Universities Science,and Technology Research Project(ZD20131017)
文摘In this paper, we focus on a constant elasticity of variance （CEV） modeland want to find its optimal strategies for a mean-variance problem under two constrainedcontrols： reinsurance/new business and investment （no-shorting）. First, aLagrange multiplier is introduced to simplify the mean-variance problem and thecorresponding Hamilton-Jacobi-Bellman （HJB） equation is established. Via a powertransformation technique and variable change method, the optimal strategies withthe Lagrange multiplier are obtained. Final, based on the Lagrange duality theorem,the optimal strategies and optimal value for the original problem （i.e., the efficientstrategies and efficient frontier） are derived explicitly.
文摘In this note, we shall prove an interesting result.Theorem. Let 2 be a piece of surface without an umbilical point in 3-dimensional constant curvature space M<sup>3</sup>（C） and possess a constant mean curvature C<sub>1</sub> （C<sub>1</sub>】0）. ∑ can be isometric to a piece of the surface ∑<sup>*</sup> without an umbilical point, ∑<sup>*</sup> owning a constant mean curvature C<sub>2</sub>（C<sub>2</sub>】0 and C<sub>1</sub>≠C<sub>2</sub>） in M<sup>3</sup>（C）
文摘In fairly good agreement with the consensus range of dark energy to matter this ratio of the critical density is suggested to be connected with the golden mean φ=0.6180339887, yielding for dark energy to matter mass fractions .?Assuming the baryonic matter to be only 4.432%, the ratio of matter to baryonic matter would be , and further the ratio of dark matter to baryonic one . If one subtracts from the dark matter a contribution of antimatter with the same mass of baryonic matter, according to the antigravity theories of Villata respectively Hajdukovic, the remaining mass ratio would yield . Replacing the “Madelung” constant α of Villata’s “lattice universe” by φ, one reaches again 1 + φas the ratio of the repulsive mass contribution to the attractive one. Assuming instead of a 3D lattice a flat 2D one of rocksalt type, the numerical similarity between the Madelung constant and φ−1 could not be just coincidence. The proposed scaling of the cosmological mass fractions with the square of the most irrational universal number φmay indicate that the chaotic cosmological processes have reached a quite stable equilibrium. This may be confirmed by another, but similar representation of the mass constituents by the Archimedes’ constant π, giving for respectively for the dark components . However, the intimate connection of φ with its reciprocal may ignite the discussion whether our universe is intertwined with another universe or even part of a multiverse with the dark constituents contributed from there.
文摘In this peper the authors give the curvature estimates for complete strongly stable surfaces with constant mean curvature in space forms and extend the related results.
基金supported in part by the NSFC(11801496,11926352)the Fok Ying-Tung Education Foundation(China)the Hubei Key Laboratory of Applied Mathematics(Hubei University).
文摘In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.
基金Project supported by the Natural Science Foundation of Anhui Province of China (Grant No. 1208085MA09)the Science Research Program of Institutions of Higher Education of Anhui Province of China (Grant No. KJ2012Z297)the Fundamental Research Funds for the Central Universities (Grant No. SWJTU12ZT11)
文摘Using a new set of nucleon coupling constants CZll the properties of a proto neutron star are examined within the framework of the relativistic mean-field theory for the baryon octet system. It is found that the relative number density of A,≡ , and ≡0 for CZll are all smaller than those for GL97 and for both CZ11 and GL97, ∑-∑0 and ∑＋ do not appear. It is also found that the pressure and the maximum mass for CZll are all smaller than those for GL97. The maximum mass for CZ11 decreases by approximately 9 percent compared with that for GL97.
文摘Sommerfeld’s fundamental fine-structure constant α once more gives reason to be amazed. This comment is a Chapter of a publication in preparation dealing mainly with golden ratio signature behind Preston Guynn’s famous matter/space approach. As a result we present a relation of α to the galactic velocity , mediated by the circle constant π, which points to an omnipresent importance of this constant and its intrinsic reciprocity pecularity: α ≈ π<sup>2</sup>|β<sub>g</sub>| respectively . The designation fine-structure constant should be replaced simply by Sommerfeld’s constant. We present golden mean-based approximations for α as well as for electron’s charge and mass and connect the word average value of interaction coupling constant α<sub>s</sub>(m<sub>z</sub>) with |β<sub>g</sub>|.
