Stability and curvature estimates for minimal graphs with flat normal bundles
نویسنده
چکیده
It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs in view of an example of Lawson and Osserman. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minimal graphs of dimension no greater than six and any codimension is flat if the the normal bundle is flat and the density at infinity is finite. Such a Bernstein type theorem holds in any dimension if we assume additionally growth conditions on the volume element.
منابع مشابه
Para-Kahler tangent bundles of constant para-holomorphic sectional curvature
We characterize the natural diagonal almost product (locally product) structures on the tangent bundle of a Riemannian manifold. We obtain the conditions under which the tangent bundle endowed with the determined structure and with a metric of natural diagonal lift type is a Riemannian almost product (locally product) manifold, or an (almost) para-Hermitian manifold. We find the natural diagona...
متن کاملRigidity of Compact Minimal Submanifolds in a Sphere
In this paper we study n-dimensional compact minimal submanifolds in S with scalar curvature S satisfying the pinching condition S > n(n − 2). We show that for p ≤ 2 these submanifolds are totally geodesic (cf. Theorem 3.2 and Corollary 3.1). However, for codimension p ≥ 2, we prove the result under an additional restrictions on the curvature tensor corresponding to the normal connection (cf. T...
متن کاملInductive Analysis on Singular Minimal Hypersurfaces
The geometric analysis of a minimal hypersurface H within some Riemannian manifold (M, g) with second fundamental form A usually involves the scalar quantity |A|2 = sum of squared principal curvatures. A few classical examples are seen from Simons type inequalities like: ∆H |A|2 ≥ −C · (1 + |A|2)2 or the stability condition (valid in particular for area minimizers): 0 ≤ Area(f) = ∫ H |∇Hf |2 − ...
متن کاملTwisting and Nonnegative Curvature Metrics on Vector Bundles over the round Sphere
A complete noncompact manifold M with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold S called the soul of M . When S is a round sphere we show that the clutching map of this bundle is restricted; this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval [0, κ]. ...
متن کاملOn the Existence of Cross Sections in Locally Flat Bundles
1. A principal bundle 33(X, G) with base space X and group G is called "locally flat" if the structural group G can be reduced to a totally disconnected subgroup GiCG, or in other words: if there exists a totally disconnected subgroup GiQG, a principal bundle 33i(X, Gi) and an injection j: Soi—>33 [l ]. For instance, if a differentiable bundle 33 admits an infinitesimal connection whose curvatu...
متن کامل