Displaying similar documents to “A pointwise inequality in submanifold theory”

On total curvature of immersions and minimal submanifolds of spheres

Giovanni Rotondaro (1993)

Commentationes Mathematicae Universitatis Carolinae

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For closed immersed submanifolds of Euclidean spaces, we prove that ∫ | μ | 2 d V ≥ V / R 2 , where μ is the mean curvature field, V the volume of the given submanifold and R is the radius of the smallest sphere enclosing the submanifold. Moreover, we prove that the equality holds only for minimal submanifolds of this sphere.

A pinching theorem on complete submanifolds with parallel mean curvature vectors

Ziqi Sun (2003)

Colloquium Mathematicae

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Let M be an n-dimensional complete immersed submanifold with parallel mean curvature vectors in an (n+p)-dimensional Riemannian manifold N of constant curvature c > 0. Denote the square of length and the length of the trace of the second fundamental tensor of M by S and H, respectively. We prove that if S ≤ 1/(n-1) H² + 2c, n ≥ 4, or S ≤ 1/2 H² + min(2,(3p-3)/(2p-3))c, n = 3, then M is umbilical. This result generalizes the...