Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H（n＋1）（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that ...Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H（n＋1）（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk（r）×H（n-k）（-1/（r2 ＋ ρ2））, where r 〉 0 and 1 〈 k 〈 n - 1;（2）if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S（n-1）（r） × H1（-1/（r2 ＋ρ2）） or S1（r） × H（n-1）（-1/（r2 ＋ρ2））,r 〉 0, if one of the following conditions is satisfied （i） S≤（n-1）t22＋c2t（-2）2 on Mn or （ii）S≥ （n-1）t21＋c2t（-2）1 on Mn or（iii）（n-1）t22＋c2t（-2）2≤ S≤（n-1）t21＋c2t（-2）1 on Mn, where t_1 and t_2 are the positive real roots of （1.5）.展开更多
In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stabil...In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.展开更多
Let M be an n（≥ 3）-dimensional completely non-compact spacelike hypersurface in the de Sitter space S1^n＋1 （1） with constant mean curvature and nonnegative sectional curvature. It is proved that M is isometric t...Let M be an n（≥ 3）-dimensional completely non-compact spacelike hypersurface in the de Sitter space S1^n＋1 （1） with constant mean curvature and nonnegative sectional curvature. It is proved that M is isometric to a hyperbolic cylinder or an Euclidean space if H ≥ 1. When 2√n-1/n〈 H 〈 1, there exists a complete rotation hypersurfaces which is not a hyperbolic cylinder.展开更多
By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant...By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1.展开更多
In this paper,the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied.It is proved that the flow converges to a unique geodesi...In this paper,the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied.It is proved that the flow converges to a unique geodesic sphere,i.e.,every principle curvature of the hypersurfaces converges to a same constant under the flow.展开更多
We give a general setting for constructing a Weierstrass representation formula for simply connected minimal surfaces in a Riemannian manifold. Then, we construct examples of minimal surfaces in the three dimensional ...We give a general setting for constructing a Weierstrass representation formula for simply connected minimal surfaces in a Riemannian manifold. Then, we construct examples of minimal surfaces in the three dimensional Heisenberg group and in the product of the hyperbolic plane with the real line.展开更多
In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with n...In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with non-negative mean curvature in the non-parametric case. We use the mean curvature flow method.展开更多
The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature.Furthermore,a second order partial differential equati...The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature.Furthermore,a second order partial differential equation for the Gauss map is obtained,and it is shown that this equation is the complete integrability condition of the representation.展开更多
基金supported by NSF of Shaanxi Province (SJ08A31)NSF of Shaanxi Educational Committee (2008JK484+1 种基金2010JK642)Talent Fund of Xi'an University of Architecture and Technology
文摘Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H（n＋1）（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk（r）×H（n-k）（-1/（r2 ＋ ρ2））, where r 〉 0 and 1 〈 k 〈 n - 1;（2）if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S（n-1）（r） × H1（-1/（r2 ＋ρ2）） or S1（r） × H（n-1）（-1/（r2 ＋ρ2））,r 〉 0, if one of the following conditions is satisfied （i） S≤（n-1）t22＋c2t（-2）2 on Mn or （ii）S≥ （n-1）t21＋c2t（-2）1 on Mn or（iii）（n-1）t22＋c2t（-2）2≤ S≤（n-1）t21＋c2t（-2）1 on Mn, where t_1 and t_2 are the positive real roots of （1.5）.
基金supported by the King Saud University D.S.F.P program
文摘In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.
基金The NNSFC (10371047) and the NSF (04KJD110192) of the Education Department of Jiangsu Province, China.
文摘Let M be an n（≥ 3）-dimensional completely non-compact spacelike hypersurface in the de Sitter space S1^n＋1 （1） with constant mean curvature and nonnegative sectional curvature. It is proved that M is isometric to a hyperbolic cylinder or an Euclidean space if H ≥ 1. When 2√n-1/n〈 H 〈 1, there exists a complete rotation hypersurfaces which is not a hyperbolic cylinder.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 1071084) the National Basic Research Project for Nonlinear Science.
文摘By using the method of integrable system, we study the deformation of constant mean curvature surfaces in three-dimensional hyperbolic space form H3. We also obtain a Weierstrass representation formula of the constant mean curvature surfaces with mean curvature greater than 1.
文摘In this paper,the motion of inverse mean curvature flow which starts from a closed star-sharped hypersurface in special rotationally symmetric spaces is studied.It is proved that the flow converges to a unique geodesic sphere,i.e.,every principle curvature of the hypersurfaces converges to a same constant under the flow.
基金Work partially supported by RAS,INdAM,FAPESP and CNPq
文摘We give a general setting for constructing a Weierstrass representation formula for simply connected minimal surfaces in a Riemannian manifold. Then, we construct examples of minimal surfaces in the three dimensional Heisenberg group and in the product of the hyperbolic plane with the real line.
文摘In this paper, we reprove a theorem of M. Anderson [Invent. Math., 69 （1982）, pp. 477-494] which established the existence of a minimal hypersurface in the hyperbolic space with prescribed asymptotic boundary with non-negative mean curvature in the non-parametric case. We use the mean curvature flow method.
基金Supported by National Natural Science Foundation of China(1097102911201400+2 种基金11026062)Project of Henan Provincial Department of Education(2011A110015)Talent Youth Teacher Fund of Xinyang Normal University
基金supported by the National Natural Science Foundation of China (Nos. 10571068,10871149)the Research Fund for the Doctoral Program of Higher Education (No. 200804860046)
文摘The authors define the Gauss map of surfaces in the three-dimensional Heisenberg group and give a representation formula for surfaces of prescribed mean curvature.Furthermore,a second order partial differential equation for the Gauss map is obtained,and it is shown that this equation is the complete integrability condition of the representation.
