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On the DDVV conjecture and the comass in calibrated geometry. I. (English) Zbl 1180.53055
Let \(M^n\) be a \(n\)-dimensional manifold isometrically immersed into the space form \(N^{n+m}(c)\) of constant sectional curvature \(c\). In the study of submanifolds theory, De Smet, Dillen, Verstraelen and Vrancken made the following (DDVV) conjecture: Let \(h\) be the second fundamental form, let \(H=\frac 1n\text{trace}\,h\) be the mean curvature tensor, let \(\rho\) and \(\rho^\perp\) be the normalized scalar curvature for the tangent and the normal bundle, resp. Then \(\rho+\rho^\perp\leq | H|^2+C\). This Conjecture, in terms of matrices, can be reformulated as follows conjecture \(P(n,m)\). Let \(A_1,\dots,A_m\) be the traceless \(n\times n\) symmetric matrices. For \(n,m\geq 2\) we have \[ \left(\sum^m_{r=1}\| A_r\|^2\right)^2\geq 2\left(\sum_{r<s} \|[A_r,A_s]\|^2\right). \] The main result of the paper is the following Theorem. Let \(A_i(i=1,\dots,m)\) be \(3\times 3\) symmetric matrices. Then \[ \left(\sum^m_{i=1}\| A_i\|^2\right)^2\geq 2 \left(\sum_{i<j}\| [A_i,A_js]\|^2\right). \] That is, the Conjecture \(P(3,m)\) for \(m\geq 2\) is true.

53C38 Calibrations and calibrated geometries
53C40 Global submanifolds
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