Calabi curves as holomorphic Legendre curves and Chen's inequality (Q2782736)

Calabi [see \textit{H. B. Lawson}, Proc. Carolina Conf. on Holomorphic Mappings and Minimal Surfaces, Chapel Hill, 1970, 86-107 (1970; Zbl 0217.47006] proved that up to a holomorphic congruence there is a unique holomorphic curve \({\mathcal C}_n\) of constant Gaussian curvature in a complex projective space \(\mathbb{C}\mathbb{P} ^n\) which does not belong to any totally geodesic complex projective space of lower dimension. Well known examples of complex contact manifolds are the odd-dimensional complex projective spaces \(\mathbb{C}\mathbb{P} ^{2m+1}\). The notion of a holomorphic Legendre curve was introduced by \textit{C. Baikoussis, D. E. Blair} and \textit{F. Gouli Andreon}, Bull. Inst. Math., Acad. Sin. 26, No.3, 179-194 (1998; Zbl 0908.53012)]. NEWLINENEWLINENEWLINEThe authors prove that the Calabi curves \({\mathcal C}_{2m+1}\) of odd index can be brought by a holomorphic congruence into the position of a holomorphic Legendre curve in \(\mathbb{C}\mathbb{P} ^{2n+1}\). The same is true for any \({\mathcal C}_{k}, k \leq n\). Applying this result, the authors prove that the curves \({\mathcal C}_{2m+1}\) give rise to totally real, minimal immersions of \(S^3\) into \(\mathbb{C}\mathbb{P} ^{2n+1}\) for which the equality case of Chen's inequality [see \textit{B.-Y. Chen}, Arch. Math. 60, 568-578 (1993; Zbl 0811.53060)] of the infimum of the sectional curvature of a submanifold holds. The authors also prove that there is no Veronese surface which is an integral submanifold of the complex contact structure on \(\mathbb{C}\mathbb{P} ^{5}\).