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  • Conformal minimal immersions of constant curvature of Riemann surfaces into symmetric spaces and flag manifolds
    Publication . Rehman, Mehmood ur; Pacheco, Rui Miguel Nobre Martins
    Following the seminal result by Eugenio Calabi establishing the local classification of complex submanifolds with constant holomorphic sectional curvature in complex space forms, several researchers have investigated minimal immersions with constant curvature of Riemann surfaces into symmetric spaces. For isometric immersions, recall that minimality is equivalent to harmonicity, hence the rich theory of harmonic maps has played here an important role. There exists a well-established theory on twistorial constructions of harmonic maps from Riemann surfaces into symmetric spaces. An important class of twistor lifts is that of primitive maps into k-symmetric spaces. In this thesis, we investigate primitive immersions of constant curvature from Riemann surfaces into flag manifolds equipped with invariant metrics and their canonical structure of k-symmetric spaces. First we consider the case of primitive lifts associated to pseudoholomorphic maps from surfaces into complex Grassmannians. We establish that any such primitive lift from the twosphere S2 into a flag manifold has constant curvature with respect to all invariant metrics, provided that it has constant curvature with respect to at least one such invariant metric. This lead us to conclude as a corollary that any primitive immersion of constant curvature from S2 into the full flag manifold is unitarily equivalent to the primitive lift of a Veronese map. We prove a partial generalization of this result to the case where the domain is a general simply connected Riemann surface. On the way, we consider the problem of finding the invariant metric on the flag manifold, under a certain normalization condition, that maximizes the induced area of the two-sphere by a given primitive immersion. Finally, we explicitly classify all the primitive immersions of constant curvature from S2 into certain low dimensional flag manifolds, namely F2,1,1 and F2,2,1.
    2025-12-05Doctoral thesis Open access Show more