A class of twisted special Lagrangian submanifolds in T＊R^n and a kind of austere submanifold from conormal bundle of minimal surface of R^3 are constructed.
In this paper,we consider the Neumann problem for special Lagrangian equations with critical phase.The global gradient and Hessian estimates are obtained.Using the method of continuity,we prove the existence of soluti...In this paper,we consider the Neumann problem for special Lagrangian equations with critical phase.The global gradient and Hessian estimates are obtained.Using the method of continuity,we prove the existence of solutions to this problem.展开更多
nspired by the Neumann problem of real special Lagrangian equations with supercritical phase,we consider the Neumann problem of complex special Lagrangian equations with supercritical phase in this paper,and establish...nspired by the Neumann problem of real special Lagrangian equations with supercritical phase,we consider the Neumann problem of complex special Lagrangian equations with supercritical phase in this paper,and establish the global C^2 estimates and the existence theorem by the method of continuity.展开更多
On the total space of the line bundle π: π*1T*P1(◎)π2*T*P1 → P1× P1, acomplete Ricci-flat Kaehler metric and a smooth special Lagrangian fibration are given.This special Lagrangian fibration is smoothly buil...On the total space of the line bundle π: π*1T*P1(◎)π2*T*P1 → P1× P1, acomplete Ricci-flat Kaehler metric and a smooth special Lagrangian fibration are given.This special Lagrangian fibration is smoothly built up of 4 Harvey-Lawson's models in 4directions.展开更多
A suffcient condition for a set of calibrated submanifolds to be area-minimizing with multiplicities,also call weighted area-minimizing under diffeomorphisms (WAMD) is stated.We construct some WAMD submanifolds by ass...A suffcient condition for a set of calibrated submanifolds to be area-minimizing with multiplicities,also call weighted area-minimizing under diffeomorphisms (WAMD) is stated.We construct some WAMD submanifolds by assembling pieces of special Lagrangian (SL) normal bundles including the one of three surfaces meeting at an angle of 120° along soap-film-like singularities.We also mention a symmetry property of SL submanifolds and Bjrling type problem for SL normal bundles.展开更多
Let SO(n) act in the standard way on Cn and extend this action in the usual way to Cn+1 =C+Cn. It is shown that a nonsingular special Lagrangian submanifold L (?) Cn+1 that is invariant under this SO(n)-action interse...Let SO(n) act in the standard way on Cn and extend this action in the usual way to Cn+1 =C+Cn. It is shown that a nonsingular special Lagrangian submanifold L (?) Cn+1 that is invariant under this SO(n)-action intersects the fixed C (?) Cn+1 in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A(?)C lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear pde and ultimately calls on the work of Gerard and Tahara to prove the existence of the extension.展开更多
Unification of fundamental forces is the dream of physics. Nevertheless, unfortunately gravitational force operators to be isolated in its geometrical content from other forces. This encourages some researchers to pro...Unification of fundamental forces is the dream of physics. Nevertheless, unfortunately gravitational force operators to be isolated in its geometrical content from other forces. This encourages some researchers to propose the so-called gravimagnetic field to unify gravity with other forces and to explain some cosmological problems at the early universe. This motivates to construct a new model to confirm the existence of gravitomagnetic and a corresponding magnetic field associated with any field. Using the formal Newton definition of force and considering the magnetic force to be related to the time varying mass, the magnetic force is shown to be equal to the centrifugal force. This equality is typical when treating a particle as string. Using also the definition of force in terms of potential and electric force only, energy is shown to be conserved. The Newton force can be defined also in terms of four-dimensional potential with the time varying part related to the magnetic potential. When the particle is treated as a string, energy conservation holds, while for ordinary particle, the Lagrangian is conserved. The energy conservation holds for special relativity also for energy per unit mass. The definition of acceleration for forces that obeys inverse square law shows also the magnetic force is equal to the centrifugal force.展开更多
We study Lagrangian submanifolds foliated by （n - 1）-spheres in R^2n for n ≥ 3. We give a general parametrization for such submanifolds, and refine that description when the submanifold is special Lagrangian, self-...We study Lagrangian submanifolds foliated by （n - 1）-spheres in R^2n for n ≥ 3. We give a general parametrization for such submanifolds, and refine that description when the submanifold is special Lagrangian, self-similar, Hamiltonian stationary or has mean curvature vector of constant length. In all these cases, the submanifold is centered, i.e. invariant under the action of SO（n）. It suffices then to solve a simple ODE in two variables to describe the geometry of the solutions.展开更多
The deformation of a compact complex Lagrangian submanifold in a hyper-Kaehler manifold and the moduli space are studied. It is proved that the moduli space Mc1 is a special Kaehler manifold, where special means that ...The deformation of a compact complex Lagrangian submanifold in a hyper-Kaehler manifold and the moduli space are studied. It is proved that the moduli space Mc1 is a special Kaehler manifold, where special means that there is a real flat torsionfree symplectic connection satisfying dI = 0 (I is a complex structure of Mcl). Thus, following [4], one knows thatT*Mcl is a hyper-Kaehler manifold and then that Mcl is a complex Lagrangian submanifold in T* Mcl.展开更多
文摘A class of twisted special Lagrangian submanifolds in T＊R^n and a kind of austere submanifold from conormal bundle of minimal surface of R^3 are constructed.
文摘In this paper,we consider the Neumann problem for special Lagrangian equations with critical phase.The global gradient and Hessian estimates are obtained.Using the method of continuity,we prove the existence of solutions to this problem.
基金supported by ZJNSF No. LY17A010022NSFC No.11771396+2 种基金supported by NSFC No. 11471188Wu Wen-Tsun Key Laboratory of Mathematics in USTCsupported by China Scholarship Council
文摘nspired by the Neumann problem of real special Lagrangian equations with supercritical phase,we consider the Neumann problem of complex special Lagrangian equations with supercritical phase in this paper,and establish the global C^2 estimates and the existence theorem by the method of continuity.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10101004).
文摘On the total space of the line bundle π: π*1T*P1(◎)π2*T*P1 → P1× P1, acomplete Ricci-flat Kaehler metric and a smooth special Lagrangian fibration are given.This special Lagrangian fibration is smoothly built up of 4 Harvey-Lawson's models in 4directions.
基金supported in part by the National Foundation for Science and Technology Development,Vietnam (Grant No.101.01.30.09)
文摘A suffcient condition for a set of calibrated submanifolds to be area-minimizing with multiplicities,also call weighted area-minimizing under diffeomorphisms (WAMD) is stated.We construct some WAMD submanifolds by assembling pieces of special Lagrangian (SL) normal bundles including the one of three surfaces meeting at an angle of 120° along soap-film-like singularities.We also mention a symmetry property of SL submanifolds and Bjrling type problem for SL normal bundles.
基金Project supported by Duke University via a research grant, the NSF via DMS-0103884the Mathematical Sciences Research Institute, and Columbia University.
文摘Let SO(n) act in the standard way on Cn and extend this action in the usual way to Cn+1 =C+Cn. It is shown that a nonsingular special Lagrangian submanifold L (?) Cn+1 that is invariant under this SO(n)-action intersects the fixed C (?) Cn+1 in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A(?)C lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear pde and ultimately calls on the work of Gerard and Tahara to prove the existence of the extension.
文摘Unification of fundamental forces is the dream of physics. Nevertheless, unfortunately gravitational force operators to be isolated in its geometrical content from other forces. This encourages some researchers to propose the so-called gravimagnetic field to unify gravity with other forces and to explain some cosmological problems at the early universe. This motivates to construct a new model to confirm the existence of gravitomagnetic and a corresponding magnetic field associated with any field. Using the formal Newton definition of force and considering the magnetic force to be related to the time varying mass, the magnetic force is shown to be equal to the centrifugal force. This equality is typical when treating a particle as string. Using also the definition of force in terms of potential and electric force only, energy is shown to be conserved. The Newton force can be defined also in terms of four-dimensional potential with the time varying part related to the magnetic potential. When the particle is treated as a string, energy conservation holds, while for ordinary particle, the Lagrangian is conserved. The energy conservation holds for special relativity also for energy per unit mass. The definition of acceleration for forces that obeys inverse square law shows also the magnetic force is equal to the centrifugal force.
文摘We study Lagrangian submanifolds foliated by （n - 1）-spheres in R^2n for n ≥ 3. We give a general parametrization for such submanifolds, and refine that description when the submanifold is special Lagrangian, self-similar, Hamiltonian stationary or has mean curvature vector of constant length. In all these cases, the submanifold is centered, i.e. invariant under the action of SO（n）. It suffices then to solve a simple ODE in two variables to describe the geometry of the solutions.
文摘The deformation of a compact complex Lagrangian submanifold in a hyper-Kaehler manifold and the moduli space are studied. It is proved that the moduli space Mc1 is a special Kaehler manifold, where special means that there is a real flat torsionfree symplectic connection satisfying dI = 0 (I is a complex structure of Mcl). Thus, following [4], one knows thatT*Mcl is a hyper-Kaehler manifold and then that Mcl is a complex Lagrangian submanifold in T* Mcl.
