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On total curvature of immersions and minimal submanifolds of spheres. (English) Zbl 0787.53049
For any smooth immersion of a closed manifold \(M\) into Euclidean space a lower bound for the total mean curvature is obtained. More precisely, it is shown that \(\int_ M H^ 2 \geq V/R^ 2\) where \(H\), \(V\), \(R\) denote the norm of the mean curvature vector, the volume, and the circumradius of \(M\), respectively. This is a straightforward consequence of the Cauchy-Schwarz inequality and the Minkowski formula for the scalar product \((u,v)\to \int_ M\langle u,v\rangle\) of vector valued functions. Equality holds if and only if \(M\) is a minimal immersion in a hypersphere of radius \(R\).
MSC:
53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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