We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc \({\mathbb D}\) in \(\mathbb C\) to the unit ball \({\mathbb B}^n\) in \(\mathbb R^n\), \(n\ge 2\), at any point where the map is conformal. In dimension \(n=2\) this generalizes the classical Schwarz-Pick lemma to harmonic maps \(\mathbb D\to\mathbb D\) which are conformal only at the reference point. In dimensions \(n\ge 3\) it gives the optimal Schwarz-Pick lemma for conformal minimal discs \(\mathbb D\to {\mathbb B}^n\). Let \({\mathcal M}\) denote the restriction of the Bergman metric on the complex \(n\)-ball to the real \(n\)-ball \({\mathbb B}^n\). We show that conformal harmonic immersions \(M \to ({\mathbb B}^n,{\mathcal M})\) from any hyperbolic open Riemann surface \(M\) with its natural Poincaré metric are distance-decreasing, and the isometries are precisely the conformal embeddings of \(\mathbb D\) onto affine discs in \({\mathbb B}^n\). (Joint work with David Kalaj.)

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H. Anciaux and K. Panagiotidou [1] initiated the study of non-degenerate real hypersurfaces in non-flat indefinite complex space forms in 2015. Next, in 2019 M. Kimura and M. Ortega [2] further developed their ideas, with a focus on Hopf real hypersurfaces in the indefinite complex projective space \(\mathbb CP^n_p\). In this work we are interested in the study of non-degenerate ruled real hypersurfaces in \(\mathbb CP^n_p\). We first define such hypersurfaces, then give basic characterizations. We also construct their parameterization. They are described as follows. Given a regular curve \(\alpha\) in \(\mathbb CP^n_p\), then the family of the complete, connected, complex \((n − 1)\)-dimensional totally geodesic submanifolds orthogonal to \(\alpha'\) and \(J\alpha'\), where \(J\) is the complex structure, generates a ruled real hypersurface. This representation agrees with the one given by M. Lohnherr and H. Reckziegel in the Riemannian case [3]. Further insights are given into the cases when the ruled real hypersurfaces are minimal or have constant sectional curvatures. The present results are part of a joint work together with prof. M. Ortega and prof. J.D. Pérez.

[1] H. Anciaux, K. Panagiotidou, Hopf Hypersurfaces in pseudo-Riemannian complex and para-complex space forms, Diff. Geom. Appl. 42 (2015) 1-14.
[2] M. Kimura, M. Ortega, Hopf Real Hypersurfaces in Indefinite Complex Projective, Mediterr. J. Math. (2019) 16:27.
[3] M. Lohnherr, H. Reckziegel, On ruled real hypersurfaces in complex space forms. Geom. Dedicata 74 (1999), no. 3, 267–286.

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