Abstract
Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson–Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.
Highlights
We consider an oriented n-dimensional submanifold M in Rn+m with n ≥ 3, m ≥ 2
We are interested in the case m ≥ 2 where the geometry of this Grassmann manifold is more complicated
By the theorem of Ruh and Vilms [16], γ is harmonic if and only if M has parallel mean curvature. This result applies in particular to the case where M is a minimal submanifold of Euclidean space
Summary
We consider an oriented n-dimensional submanifold M in Rn+m with n ≥ 3, m ≥ 2. Hildebrandt et al [10] started a systematic approach on the basis of the aforementioned Ruh–Vilms theorem That is, they developed and employed the theory of harmonic maps and the convex geometry of Grassmannian manifolds, and obtained Bernstein type results in general dimension and codimension. We shall use the iteration method of [10], we can explore the relation with curvature estimates, and we shall utilize a version of the telescoping trick (Theorem 4.1) to obtain a quantitatively controlled Gauss image shrinking process (Theorems 5.1 and 6.1) In this way, we can understand why the submanifold is flat as the Bernstein result asserts. Somewhat more refined results can be obtained, as will be pointed out in the final remarks of this paper
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