共查询到20条相似文献,搜索用时 31 毫秒
1.
Shurong Sun Yuming Shi Shaozhu Chen 《Journal of Mathematical Analysis and Applications》2007,327(2):1360-1380
In this paper, the Glazman-Krein-Naimark theory for a class of discrete Hamiltonian systems is developed. A minimal and a maximal operators, GKN-sets, and a boundary space for the system are introduced. Algebraic characterizations of the domains of self-adjoint extensions of the minimal operator are given. A close relationship between the domains of self-adjoint extensions and the GKN-sets is established. It is shown that there exist one-to-one correspondences among the set of all the self-adjoint extensions, the set of all the d-dimensional Lagrangian subspaces of the boundary space, and the set of all the complete Lagrangian subspaces of the boundary space. 相似文献
2.
ZHOU Min 《中国科学 数学(英文版)》2014,57(5):1033-1044
We establish the Mather theory for a type of piecewise smooth and positive definite Lagrangian systems.It models a mechanical system subject to external impulsive forcing.We show the existence of the minimal measure and the Lipschitz property of Aubry set.In addition,the weak KAM solution to this kind of piecewise smooth Lagrangian is also established. 相似文献
3.
It is shown that for every non-negative integer n, there isa real n-dimensional family of minimal Lagrangian tori in CP2,and hence of special Lagrangian cones in C3 whose link is atorus. The proof utilises the fact that such tori arise fromintegrable systems, and can be described using algebro-geometric(spectral curve) data. 相似文献
4.
Fang Wang 《Journal of Differential Equations》2009,247(12):3258-3282
5.
Barbara Opozda 《Monatshefte für Mathematik》2009,156(4):357-370
One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant
curvature in C
2 must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection.
In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds.
In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C
2. Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian)
geometry.
The research supported by the KBN grant 1 PO3A 034 26. 相似文献
6.
Barbara Opozda 《Monatshefte für Mathematik》2009,145(1):357-370
One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant
curvature in C
2 must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection.
In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds.
In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C
2. Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian)
geometry. 相似文献
7.
Li Haizhong Ma Hui Van der Veken Joeri Vrancken Luc Wang Xianfeng 《中国科学 数学(英文版)》2020,63(8):1441-1462
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. 相似文献
8.
In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel. 相似文献
9.
In this paper, an estimate of the constant scalar curvature of a compact non- minimal pseudo-umbilical Lagrangian submanifold in CP3 is obtained. As its application, we prove that compact Einstein pseudo-umbilical Lagrangian submanifolds in CP3 must be minimal. 相似文献
10.
Knut Smoczyk 《Mathematische Zeitschrift》2002,240(4):849-883
We prove that symplectic maps between Riemann surfaces L, M of constant, nonpositive and equal curvature converge to minimal symplectic maps, if the Lagrangian angle for the corresponding Lagrangian submanifold in the cross product space satisfies . If one considers a 4-dimensional K?hler-Einstein manifold of nonpositive scalar curvature that admits two complex structures J, K which commute and assumes that is a compact oriented Lagrangian submanifold w.r.t. J such that the K?hler form w.r.t.K restricted to L is positive and , then L converges under the mean curvature flow to a minimal Lagrangian submanifold which is calibrated w.r.t. .
Received: 11 April 2001 / Published online: 29 April 2002 相似文献
11.
A. A. Kazhymurat 《Siberian Mathematical Journal》2018,59(4):641-647
Under study is the energy functional on the set of Lagrangian tori in the complex projective plane. We prove that the value of the energy functional for a certain family of Hamiltonian minimal Lagrangian tori in the complex projective plane is strictly larger than for the Clifford torus. 相似文献
12.
A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane \({\mathbb C}P^2\). Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional \(W^{-}\) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori. 相似文献
13.
DONG Yuxin & LU Guozhen Institute of Mathematics Fudan University Shanghai China Department of Mathematics Wayne State University Detroit MI 《中国科学A辑(英文版)》2005,48(11):1505-1516
In this paper, we determine all second order minimal Lagrangian submanifolds in complex space forms whose cubic forms have the largest non-trivial continuous symmetries. We describe these minimal Lagrangian submanifolds from the viewpoint of Bryant and study their geometric properties. 相似文献
14.
We prove that for one cannot immerse as a minimal Lagrangian manifold into a hyperK?hler manifold. More generally we show that any minimal Lagrangian immersion
of an orientable closed manifold into a hyperK?hler manifold must have nonvanishing second Betti number and that if , is a K?hler manifold and more precisely a K?hler submanifold in w.r.t. one of the complex structures on . In addition we derive a result for the other Betti numbers.
Received February 10, 1999 / Accepted April 23, 1999 相似文献
15.
Tommaso Pacini 《Transactions of the American Mathematical Society》2003,355(8):3343-3357
Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: e.g., finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for example, proves that the minimal Lagrangian orbits are isolated.
16.
17.
Ildefonso Castro Francisco Torralbo Francisco Urbano 《Mathematische Zeitschrift》2012,271(1-2):257-270
Hamiltonian stationary Lagrangian spheres in K?hler-Einstein surfaces are minimal. We prove that in the family of non-Einstein K?hler surfaces given by the product Σ1?×?Σ2 of two complete orientable Riemannian surfaces of different constant Gauss curvatures, there is only a (non minimal) Hamiltonian stationary Lagrangian sphere. This example, defined when the surfaces Σ1 and Σ2 are spheres, is unstable. 相似文献
18.
It is known that all weakly conformal Hamiltonian stationary Lagrangian immersions of tori in ${{\mathbb {CP}}^2}$ may be constructed by methods from integrable systems theory. This article describes the precise details of a construction which leads to a form of classification. The immersion is encoded as spectral data in a similar manner to the case of minimal Lagrangian tori in ${{\mathbb {CP}}^2}$ , but the details require a careful treatment of both the ??dressing construction?? and the spectral data to deal with a loop of flat connexions which is quadratic in the loop parameter. 相似文献
19.
From the existence of parallel spinor fields on Calabi-Yau, hyper-Kähler or complex flat manifolds, we deduce the existence of harmonic differential forms of different degrees on their minimal Lagrangian submanifolds. In particular, when the submanifolds are compact, we obtain sharp estimates on their Betti numbers which generalize those obtained by Smoczyk in [49]. When the ambient manifold is Kähler-Einstein with positive scalar curvature, and especially if it is a complex contact manifold or the complex projective space, we prove the existence of Kählerian Killing spinor fields for some particular spin c structures. Using these fields, we construct eigenforms for the Hodge Laplacian on certain minimal Lagrangian submanifolds and give some estimates for their spectra. These results also generalize some theorems by Smoczyk in [50]. Finally, applications on the Morse index of minimal Lagrangian submanifolds are obtained. 相似文献
20.
In this paper, we find some new explicit examples of Hamiltonian minimal Lagrangian submanifolds among the Lagrangian isometric immersions of a real space form in a complex space form. 相似文献