Let x:M → Rn be an umbilical free hypersurface with non-zero principal curvatures.Then x is associated with a Laguerre metric g,a Laguerre tensor L,a Laguerre form C,and a Laguerre second fundamental form B,which are...Let x:M → Rn be an umbilical free hypersurface with non-zero principal curvatures.Then x is associated with a Laguerre metric g,a Laguerre tensor L,a Laguerre form C,and a Laguerre second fundamental form B,which are invariants of x under Laguerre transformation group.An eigenvalue of Laguerre tensor L of x is called a Laguerre eigenvalue of x.In this paper,we classify all oriented hypersurfaces with constant Laguerre eigenvalues and vanishing Laguerre form.展开更多
Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is ...Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.展开更多
In this paper,we study oriented surfaces of R3 in the context of Laguerre geometry.We construct Laguerre invariants on the non-Dupin developable surfaces,which determine the surfaces up to a Laguerre transformation.Fi...In this paper,we study oriented surfaces of R3 in the context of Laguerre geometry.We construct Laguerre invariants on the non-Dupin developable surfaces,which determine the surfaces up to a Laguerre transformation.Finally,we classify the Laguerre homogeneous surfaces in R3 under the Laguerre transformation groups.展开更多
Let x ： M →R^n be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B wh...Let x ： M →R^n be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R^4.展开更多
In this paper, a self-mode-locked Nd:YVO_4 picosecond vortex laser is demonstrated, which can operate on the different Laguerre-Gaussian(LG) modes at 1 064 nm. A π/2 mode converter is utilized to realize the picoseco...In this paper, a self-mode-locked Nd:YVO_4 picosecond vortex laser is demonstrated, which can operate on the different Laguerre-Gaussian(LG) modes at 1 064 nm. A π/2 mode converter is utilized to realize the picosecond vortex laser with LG mode transformed from the high-order Hermite-Gaussian(HG) mode. For the proposed laser, the mode-locked pulse repetition rate is 1.81 GHz. The average output powers of LG_(12) mode and LG_(02) mode are 1.241 W and 1.27 W, respectively, and their slope efficiencies are 23.2% and 24%, respectively.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.10801006,10971110,10771005)
文摘Let x:M → Rn be an umbilical free hypersurface with non-zero principal curvatures.Then x is associated with a Laguerre metric g,a Laguerre tensor L,a Laguerre form C,and a Laguerre second fundamental form B,which are invariants of x under Laguerre transformation group.An eigenvalue of Laguerre tensor L of x is called a Laguerre eigenvalue of x.In this paper,we classify all oriented hypersurfaces with constant Laguerre eigenvalues and vanishing Laguerre form.
文摘Let f ： M → R3 be an oriented surface with non-degenerate second fundamental form. We denote by H and K its mean curvature and Gauss curvature. Then the Laguerre volume of f, defined by L（f） = f（H2 - K）/KdM, is an invariant under the Laguerre transformations. The critical surfaces of the functional L are called Laguerre minimal surfaces. In this paper we study the Laguerre minimal surfaces in R^3 by using the Laguerre Gauss map. It is known that a generic Laguerre minimal surface has a dual Laguerre minimal surface with the same Gauss map. In this paper we show that any surface which is not Laguerre minimal is uniquely determined by its Laguerre Gauss map. We show also that round spheres are the only compact Laguerre minimal surfaces in R^3. And we give a classification theorem of surfaces in R^3 with vanishing Laguerre form.
基金supported by National Natural Science Foundation of China (Grant No.10801006)
文摘In this paper,we study oriented surfaces of R3 in the context of Laguerre geometry.We construct Laguerre invariants on the non-Dupin developable surfaces,which determine the surfaces up to a Laguerre transformation.Finally,we classify the Laguerre homogeneous surfaces in R3 under the Laguerre transformation groups.
基金supported by National Natural Science Foundation of China (Grant No. 10801006)supported by National Natural Science Foundation of China (Grant No. 10871218)
文摘Let x ： M →R^n be an umbilical free hypersurface with non-zero principal curvatures. Then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and a Laguerre second fundamental form B which are invariants of x under Laguerre transformation group. A hypersurface x is called Laguerre isoparametric if its Laguerre form vanishes and the eigenvalues of B are constant. In this paper, we classify all Laguerre isoparametric hypersurfaces in R^4.
基金supported by the National Natural Science Foundation of China(No.61108021)the Fundamental Research Funds for the Central Universities(Nos.2013JBM091 and S16JB00010)
文摘In this paper, a self-mode-locked Nd:YVO_4 picosecond vortex laser is demonstrated, which can operate on the different Laguerre-Gaussian(LG) modes at 1 064 nm. A π/2 mode converter is utilized to realize the picosecond vortex laser with LG mode transformed from the high-order Hermite-Gaussian(HG) mode. For the proposed laser, the mode-locked pulse repetition rate is 1.81 GHz. The average output powers of LG_(12) mode and LG_(02) mode are 1.241 W and 1.27 W, respectively, and their slope efficiencies are 23.2% and 24%, respectively.
