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
Some properties of the pseudo umbilical surface M in R 4 are discussed and thus the lower for the tolal mean curvature of M is estimated. On the basis of the estimation and by using the Gauss map of M...Some properties of the pseudo umbilical surface M in R 4 are discussed and thus the lower for the tolal mean curvature of M is estimated. On the basis of the estimation and by using the Gauss map of M , a sufficient condition is given for M as a flat torus in R 4 .展开更多
Motivated by the result of Chen-Liu-Ru[1],we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(n) with ramification,which can be seen as a...Motivated by the result of Chen-Liu-Ru[1],we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(n) with ramification,which can be seen as a generalization of the results in the case of the minimal surfaces.In addition,we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.展开更多
In this paper,we give some rigidity results for complete self-shrinking surfaces properly immersed in R^(4) under some assumptions regarding their Gauss images.More precisely,we prove that this has to be a plane,provi...In this paper,we give some rigidity results for complete self-shrinking surfaces properly immersed in R^(4) under some assumptions regarding their Gauss images.More precisely,we prove that this has to be a plane,provided that the images of either Gauss map projection lies in an open hemisphere or S^(2)(1/2–√)∖S^(-1)+(1/2–√).We also give the classification of complete self-shrinking surfaces properly immersed in R^(4) provided that the images of Gauss map projection lies in some closed hemispheres.As an application of the above results,we give a new proof for the result of Zhou.Moreover,we establish a Bernstein-type theorem.展开更多
In this paper, we investigate the surfaces of revolution under the condition FIl（G） = k（G ＋ C）, where r11 is one of the Christoffel-like operators, G is the Gauss map of the surface, k is a non-constant function ...In this paper, we investigate the surfaces of revolution under the condition FIl（G） = k（G ＋ C）, where r11 is one of the Christoffel-like operators, G is the Gauss map of the surface, k is a non-constant function and C is a constant vector in Minkowski 3-space.展开更多
In this paper, the author uses Gauss map to study the topology, volume and diameter of submanifolds in a sphere. It is proved that if there exist ε, 1≥ε > 0 and a fixed unit simple p-vector a such that the Gauss...In this paper, the author uses Gauss map to study the topology, volume and diameter of submanifolds in a sphere. It is proved that if there exist ε, 1≥ε > 0 and a fixed unit simple p-vector a such that the Gauss map g of an n-dimensional complete and connected submanifold M in Sn+p satisfies (g, a) ≥ε, then M is diffeomorphic to Sn, and the volume and diameter of M satisfy εnvol(Sn) ≤vol(M) ≤ vol(Sn)/ε and επ ≤diam(M) ≤ π/ε, respectively. The author also characterizes the case where these inequalities become equalities. As an application, a differential sphere theorem for compact submanifolds in a sphere is obtained.展开更多
The relationship between CR submanifolds in a sphere and their Gauss maps are investigated.Let V be the image of a sphere by a rational holomorphic map F with degree two in another sphere.It is show that the Gauss map...The relationship between CR submanifolds in a sphere and their Gauss maps are investigated.Let V be the image of a sphere by a rational holomorphic map F with degree two in another sphere.It is show that the Gauss map of V is degenerate if and only if F is linear fractional.展开更多
The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the...The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric.Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.展开更多
In this paper we study surfaces in S^4 and their twistor Gauss maps.Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given.We find many examples of nonisotropic harmonic maps from a ...In this paper we study surfaces in S^4 and their twistor Gauss maps.Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given.We find many examples of nonisotropic harmonic maps from a surface to(?)P^3.展开更多
This paper studies the Gauss map of submanifolds in space forms defined by Willmore andSaleemi. By using Morse functions,it is proved that the degree of Gauss map is the Eulernumber of the submanifold.The tight immers...This paper studies the Gauss map of submanifolds in space forms defined by Willmore andSaleemi. By using Morse functions,it is proved that the degree of Gauss map is the Eulernumber of the submanifold.The tight immersions are also studied.展开更多
Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a cer...Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space.展开更多
Gauss maps of oriented timelike 2-surfaces in are characterized, and it is shown that Gallss maps can determine surfaces locally as they do in case. Moreover, some essential differences are discovered between the prop...Gauss maps of oriented timelike 2-surfaces in are characterized, and it is shown that Gallss maps can determine surfaces locally as they do in case. Moreover, some essential differences are discovered between the properties of the Gauss maps of surfaces in Rn and those of the Gauss maps of timelike surfaces in. In particular, a counterexample shows that a nonminimal timelike surface in cannot be essentially determined by its Gauss map.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
Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensiona...Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.展开更多
We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angl...We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.展开更多
Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,...Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,and so is the Cartan-Munzner polynomial F(x).This action is extended to a Hamiltonian action on C^(2 l).We give a new description of F(x)by the moment mapμ:C2 l→t^(*),where t≌o(m+1)is the Lie algebra of Spin(m+1).It also induces a Hamiltonian action on CP^(2 l-1).We consider the Gauss map g of N into the complex hyperquadric Q_(2 l-2)(C)■CP^(2 l-1),and show that g(N)lies in the zero level set of the moment map restricted to Q_(2 l-2)(C).展开更多
Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homoto...Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.展开更多
Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow ha...Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the v-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the v-function. Under such restrictions, curvature estimates in terms of v-function composed with the Gauss map can be carried out.展开更多
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.
文摘Some properties of the pseudo umbilical surface M in R 4 are discussed and thus the lower for the tolal mean curvature of M is estimated. On the basis of the estimation and by using the Gauss map of M , a sufficient condition is given for M as a flat torus in R 4 .
基金supported by the Fundamental Research Funds for the Central Universities(500421360)supported by NNSF of China(11571049,12071047)+1 种基金supported by NNSF of China(11971182)NSF of Fujian Province of China(2019J01066)。
文摘Motivated by the result of Chen-Liu-Ru[1],we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(n) with ramification,which can be seen as a generalization of the results in the case of the minimal surfaces.In addition,we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.
基金supported by the National Natural Science Foundation of China(11001130,11871275)the Fundamental Research Funds for the Central Universities(30917011335).
文摘In this paper,we give some rigidity results for complete self-shrinking surfaces properly immersed in R^(4) under some assumptions regarding their Gauss images.More precisely,we prove that this has to be a plane,provided that the images of either Gauss map projection lies in an open hemisphere or S^(2)(1/2–√)∖S^(-1)+(1/2–√).We also give the classification of complete self-shrinking surfaces properly immersed in R^(4) provided that the images of Gauss map projection lies in some closed hemispheres.As an application of the above results,we give a new proof for the result of Zhou.Moreover,we establish a Bernstein-type theorem.
文摘In this paper, we investigate the surfaces of revolution under the condition FIl（G） = k（G ＋ C）, where r11 is one of the Christoffel-like operators, G is the Gauss map of the surface, k is a non-constant function and C is a constant vector in Minkowski 3-space.
基金Project supported by the Fund of the Education Department of Zhejiang Province of China (No.20030707).
文摘In this paper, the author uses Gauss map to study the topology, volume and diameter of submanifolds in a sphere. It is proved that if there exist ε, 1≥ε > 0 and a fixed unit simple p-vector a such that the Gauss map g of an n-dimensional complete and connected submanifold M in Sn+p satisfies (g, a) ≥ε, then M is diffeomorphic to Sn, and the volume and diameter of M satisfy εnvol(Sn) ≤vol(M) ≤ vol(Sn)/ε and επ ≤diam(M) ≤ π/ε, respectively. The author also characterizes the case where these inequalities become equalities. As an application, a differential sphere theorem for compact submanifolds in a sphere is obtained.
文摘The relationship between CR submanifolds in a sphere and their Gauss maps are investigated.Let V be the image of a sphere by a rational holomorphic map F with degree two in another sphere.It is show that the Gauss map of V is degenerate if and only if F is linear fractional.
基金Project supported by the 973 Project of the Ministry of Science and Technology of China and the Science Foundation of the Ministry of Education of China.
文摘The author obtains a Weierstrass representation for surfaces with prescribed normal Gauss map and Gauss curvature in H3. A differential equation about the hyperbolic Gauss map is also obtained, which characterizes the relation among the hyperbolic Gauss map, the normal Gauss map and Gauss curvature. The author discusses the harmonicity of the normal Gauss map and the hyperbolic Gauss map from surface with constant Gauss curvature in H3 to S2 with certain altered conformal metric.Finally, the author considers the surface whose normal Gauss map is conformal and derives a completely nonlinear differential equation of second order which graph must satisfy.
基金Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.
文摘In this paper we study surfaces in S^4 and their twistor Gauss maps.Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given.We find many examples of nonisotropic harmonic maps from a surface to(?)P^3.
文摘This paper studies the Gauss map of submanifolds in space forms defined by Willmore andSaleemi. By using Morse functions,it is proved that the degree of Gauss map is the Eulernumber of the submanifold.The tight immersions are also studied.
基金supported by JSPS KAKENHI (Grant Nos. JP18K03265 and JP19K03461)。
文摘Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space.
文摘Gauss maps of oriented timelike 2-surfaces in are characterized, and it is shown that Gallss maps can determine surfaces locally as they do in case. Moreover, some essential differences are discovered between the properties of the Gauss maps of surfaces in Rn and those of the Gauss maps of timelike surfaces in. In particular, a counterexample shows that a nonminimal timelike surface in cannot be essentially determined by its Gauss map.
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.
文摘Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.
基金supported by the Tsinghua University-KU Leuven Bilateral Scientific Cooperation Fundcollaboration project funded by National Natural Science Foundation of China+6 种基金supported by National Natural Science Foundation of China(Grant Nos.11831005 and 11671224)supported byNational Natural Science Foundation of China(Grant Nos.11831005 and 11671223)supported by National Natural Science Foundation of China(Grant No.11571185)the Research Foundation Flanders(Grant No.11961131001)supported by the Excellence of Science Project of the Belgian Government(Grant No.GOH4518N)supported by the KU Leuven Research Fund(Grant No.3E160361)the Fundamental Research Funds for the Central Universities。
文摘We introduce a structural approach to study Lagrangian submanifolds of the complex hyperquadric in arbitrary dimension by using its family of non-integrable almost product structures.In particular,we define local angle functions encoding the geometry of the Lagrangian submanifold at hand.We prove that these functions are constant in the special case that the Lagrangian immersion is the Gauss map of an isoparametric hypersurface of a sphere and give the relation with the constant principal curvatures of the hypersurface.We also use our techniques to classify all minimal Lagrangian submanifolds of the complex hyperquadric which have constant sectional curvatures and all minimal Lagrangian submanifolds for which all local angle functions,respectively all but one,coincide.
基金supported by Japan Society for the Promotion of Science(Grant No.15H03616)。
文摘Associated with a Clifford system on R^(2 l),a Spin(m+1)action is induced on R^(2 l).An isoparametric hypersurface N in S^(2 l-1)of OT-FKM(Ozeki,Takeuchi,Ferns,Karcher and Miinzner)type is invariant under this action,and so is the Cartan-Munzner polynomial F(x).This action is extended to a Hamiltonian action on C^(2 l).We give a new description of F(x)by the moment mapμ:C2 l→t^(*),where t≌o(m+1)is the Lie algebra of Spin(m+1).It also induces a Hamiltonian action on CP^(2 l-1).We consider the Gauss map g of N into the complex hyperquadric Q_(2 l-2)(C)■CP^(2 l-1),and show that g(N)lies in the zero level set of the moment map restricted to Q_(2 l-2)(C).
基金supported by the Natural Science Foundation of China(No.11361073)supported by the Natural Science Foundation of Guangxi Province of China(No.2011GXNSFA018127)
文摘Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.
基金Project supported by the National Natural Science Foundation of China and the Science Foundation of the Ministry of Education of China
文摘Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the v-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the v-function. Under such restrictions, curvature estimates in terms of v-function composed with the Gauss map can be carried out.
