The rigidity of spacelike hypersurface Mn immersed in locally symmetric space M1n+1 is investigated,where the(normalized)scalar curvature R and mean curvature H of Mn satisfy R=aH+b,and a,b are real constants.First,an...The rigidity of spacelike hypersurface Mn immersed in locally symmetric space M1n+1 is investigated,where the(normalized)scalar curvature R and mean curvature H of Mn satisfy R=aH+b,and a,b are real constants.First,an estimate of the upper bound of the function L(nH)is given,where L is a second-order differential operator.Then,under the assumption that the square norm of the second fundamental form is bounded by a given positive constant,it is proved that Mn must be either totally umbilical or contain two distinct principle curvatures,one of which is simple.Moreover,a similar result is obtained for complete noncompact spacelike hypersurfaces in locally symmetric Einstein spacetime.Hence,some known rigidity results for hypersurface with constant scalar curvature are extended for the linear Weingarten case.展开更多
In this paper,we consider λ-translating solitons in R^(3).These surfaces are critical points of the weighted area when the density is a coordinate function.If λ=0,these surfaces evolve by translations along the mean...In this paper,we consider λ-translating solitons in R^(3).These surfaces are critical points of the weighted area when the density is a coordinate function.If λ=0,these surfaces evolve by translations along the mean curvature fow.We give a full classifcation ofλ-translating solitons that satisfy a linear Weingarten relation between their curvatures.These surfaces are planes,circular cylinders,grim reapers and certain types of cylindrical surfaces.We also prove that planes and circular cylinders are the only λ-translating soliton with constant squared norm of the second fundamental form.展开更多
基金The Natural Science Foundation of Jiangsu Province(No.BK20161412)the Fundamental Research Funds for the Central Universitiesthe Scientific Innovation Research of College Graduates in Jiangsu Province(No.KYCX17_0041)
文摘The rigidity of spacelike hypersurface Mn immersed in locally symmetric space M1n+1 is investigated,where the(normalized)scalar curvature R and mean curvature H of Mn satisfy R=aH+b,and a,b are real constants.First,an estimate of the upper bound of the function L(nH)is given,where L is a second-order differential operator.Then,under the assumption that the square norm of the second fundamental form is bounded by a given positive constant,it is proved that Mn must be either totally umbilical or contain two distinct principle curvatures,one of which is simple.Moreover,a similar result is obtained for complete noncompact spacelike hypersurfaces in locally symmetric Einstein spacetime.Hence,some known rigidity results for hypersurface with constant scalar curvature are extended for the linear Weingarten case.
基金Supported by LiaoNing Revitalization Talents Program(Grant No.XLYC2203194)the key Project for Department of Liaoning Education(Grant No.LJ112410173072)+1 种基金partially supported by MINECO/MICINN/FEDER(Grant No.PID2023-150727NB-I00)by the"María de Maeztu"Excellence Unit IMAG,reference CEX2020-001105-M,funded by MCINN/AEI/10.13039/501100011033/CEX2020-001105-M。
文摘In this paper,we consider λ-translating solitons in R^(3).These surfaces are critical points of the weighted area when the density is a coordinate function.If λ=0,these surfaces evolve by translations along the mean curvature fow.We give a full classifcation ofλ-translating solitons that satisfy a linear Weingarten relation between their curvatures.These surfaces are planes,circular cylinders,grim reapers and certain types of cylindrical surfaces.We also prove that planes and circular cylinders are the only λ-translating soliton with constant squared norm of the second fundamental form.
