We study the stability of unduloids with free boundary in the domain B between two parallel hyperplanes in R^(n+1). If the unduloid has one half of period in B and is sufficiently close to a cylinder, then for 2≤n≤1...We study the stability of unduloids with free boundary in the domain B between two parallel hyperplanes in R^(n+1). If the unduloid has one half of period in B and is sufficiently close to a cylinder, then for 2≤n≤10, it is unstable; while for n≥11, it is stable. If the unduloid has two or more halves of period in B and is sufficiently close to a cylinder, then for all n≥2, it is unstable.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
In this paper,we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold.We give C^(0),C^(1),C^(2) estimates of the flow.Using these fac...In this paper,we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold.We give C^(0),C^(1),C^(2) estimates of the flow.Using these facts,we prove that the solution exists for all time and the principal curvatures converge to 1 polynomially fast.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 11271214 and 11671224)Ben Andrews’ ARC Laureate Fellowship (Grant No. FL150100126)
文摘We study the stability of unduloids with free boundary in the domain B between two parallel hyperplanes in R^(n+1). If the unduloid has one half of period in B and is sufficiently close to a cylinder, then for 2≤n≤10, it is unstable; while for n≥11, it is stable. If the unduloid has two or more halves of period in B and is sufficiently close to a cylinder, then for all n≥2, it is unstable.
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 No.11831005)a collaboration project funded by National Natural Science Foundation of China and the Research Foundation Flanders(Grant No.11961131001)。
文摘In this paper,we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold.We give C^(0),C^(1),C^(2) estimates of the flow.Using these facts,we prove that the solution exists for all time and the principal curvatures converge to 1 polynomially fast.
