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
Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of m...Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of minimal surface （neither holomorphic nor anti-holomorphic） with constant curvature in Q2 is part of a flat totally real torus. Finally, we prove that totally real minimal fiat tori in Q2 must be totally geodesic, and we classify all the totally geodesic closed surfaces in Q2.展开更多
In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associate...In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f:S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4.Excluding the case K = 4,conformal minimal immersion f:S2 → Q2 with Gaussian curvature K2 must be totally geodesic with(K,θ) ∈ {(2,0),(2,π/2),(2,π)}.展开更多
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 National Natural Science Foundation of China(Grant Nos.11071248 and 11226079)Program of Natural Science Research of Jiangsu Higher Education Institutions of China(Grant No.12KJD110004)
文摘Our purpose is to study the minimal tori in the hyperquadric Q2. Firstly, we obtain a necessary and sufficient condition for the minimal surface in Qn which is also minimal in CPn＋1. Next, we show that this kind of minimal surface （neither holomorphic nor anti-holomorphic） with constant curvature in Q2 is part of a flat totally real torus. Finally, we prove that totally real minimal fiat tori in Q2 must be totally geodesic, and we classify all the totally geodesic closed surfaces in Q2.
基金supported by National Natural Science Foundation of China (Grant No.11071248)Knowledge Innovation Funds of CAS (Grant No.KJCX3-SYW-S03)the President Fund of GUCAS
文摘In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f:S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4.Excluding the case K = 4,conformal minimal immersion f:S2 → Q2 with Gaussian curvature K2 must be totally geodesic with(K,θ) ∈ {(2,0),(2,π/2),(2,π)}.
