Geometry Of Lagrangian Submanifolds And Related Problems

In this thesis, we mainly study geometry of Lagrangian submanifolds and related problems, including Bernstein type theorems, construction of special Lagrangian submanifolds, Hamiltonian minimal Lagrangian submanifolds and Lagrangian submanifolds with conformal Maslov form in complex space form.The celebrated theorem of Bernstein says that the only entire minimal graphs in Euclidean 3-space are planes. This result has been generalized to R⁾(n+1)), for n≤7 and general dimension under various growth condition, see [17] and the reference therein for codimension one case. For higher codimension, the situation becomes more complicated. Due to the counterexample of Lawson-Osserman [31], the higher codimension Bernstein type result is not expected to be true in the most generality. Hence we have to consider the additional suitable conditions to establish a Bernstein type result for higher codimension. In recent years, remarkable progress has been made by [23], [24], [39], [41] and [43] in Bernstein type problems of minimal submanifolds with higher codimension and special Lagrangian submanifolds. The key idea in these papers is to find a suitable subharmonic function, whose vanishing implies the minimal graph is totally geodesic. We get Bernstein type theorems of n-dimension Lagrangian submanifolds in Quaternion space Hⁿ (?) R⁴ⁿ by using the the similar idea.We know that examples of special Lagrangian submanifolds are very important for studying these submanifolds. In recent years, some examples of these submanifolds had been constructed by many people. For example, R. Harvey and H. B. Lawson in [20] gave some examples of special Lagrangian sumbmanifolds in Cⁿ, and especially they constructed a kind of special Lagrangian submanifolds by using normal bundles. And D. Joyce in [25], [26], [27]and [28] constructed explicit examples of special Lagrangian submanifolds in Cⁿ. And A.Borisenko in [1], constructed a kind of twisted special Lagrangian submanifolds in T^*R³ from the twisted normal bundle of minimal surfaces given in R³. This is a generalization of normal bundle given in [20] with dimension 3. R.L.Bryant in [3] also gave a kind of twisted special Lagrangian submanifolds in C³, this kind of submanifolds are different from the twisted special Lagrangian submanifolds given in [1]. From the constructions of twisted normal bundle of minimal surface in Rⁿ, we get a lot of examples of special Lagrangian submanifolds.Besides special Lagrangian submanifolds in Kahler manifold, the generalization of these kind of minimal Lagrangian submanifolds have received much attention recently. In [33] and [34], a nice generalization of minimal Lagrangian submanifolds called Hamiltonian-minimal submanifold was introduced and investigated by Y.G. Oh. And he also investigate this kind of submanifolds. In [14],[16],[21],[22],[9],[30] and [32], the authors made some efforts to construct examples of Hamiltonian minimal Lagrangian submanifolds in Kahler manifolds, especially in complex space forms, by means of symmetry reduction or integrable systems. There is another generalization which is Lagrangian submanifold with conformal Maslov form. A typical example of this kind of submanifold is Whitney sphere which was studied in [42],[7] and [11]. A.Ros and F.Urbano in [37] investigated Lagrangian submanifolds with conformal Maslov form in Cⁿ in general case, X.L. Chao and X.Y. Dong in [12] also investigated this kind of submanifolds, and they find a rigidity theorem for these submanifolds.In [5], the authors developed an effective method called twistor product decomposition to construct Lagrangian isometric immersions of a real-space-form Mⁿ(c) into a complex-space-forms Mⁿ(4c). Later, Y.M. Oh [35] followed their method to construct a lot of examples of such kind of Lagrangian isometric immersions. We get a lot of examples of Hamiltonian minimal Lagrangian submanifolds and Lagrangian submanifolds with conformal Maslov form in complex space forms.