Studies on Bernstein type theorems for minimal submanifolds

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Let G be a compact connected Lie group, Φ be an orthonormal representation of G on R^n with codimension 2 principal orbit type. Such (G, Φ, R^n) were classified completely by Hsiang-Lawson (1971) and they are exactly those isotropy representations of symmetric spaces of rank 2.The orbit space R^n/G is a domain of R^2, it coinsides with the Weyl chamber R^2/W (W=Weyl group) of some symmetric space G/K.Hsiang (1982) reduced the problem of constructing hypersurfaces of constant mean curvature in R^n to solving the ordinaly differential equations of curves in R^2/W, and get many examples.In this research, using Hsiang's idea, we reduced the problem of constructing complete minimal submanifolds with flat normal connection in the Euclidean spaces, to solving some ordinaly differential equations of curves in R^2/W×R.The point is that submanifolds generated from rotating curves have always flat normal connection.Theorem 1 There are many codimension 2 irreducible complete minimal submanifolds with flat normal connection in the Euclidean spaces. Those examples are diffeomorphic to the following manifolds :S^P×S^q×R, SU (2)/T^2×R, G_2/T^2×R, F_4/Spin (8)×R,....In 2000 using similar idea we get the following theorem.Theorem 2 There are many codimension 2 irreducible complete minimal submanifolds with flat normal connection in the hyperbolic spaces. Those examples are diffeomorphic to S^p×S^q×R.