Welcome to the Geometry Seminar of the Deparment of Geometry and Topology of the University of Granada. Here you can find information about the talks and events organized by the department.

# Schwarz-Pick lemma for harmonic maps which are conformal at a point

## Franc Forstnerič University of Ljubljana

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.)

Sala EINSTEIN UGR (virtual)

# Mean Curvature Flow with Boundary

## Brian White Stanford University

Sala EINSTEIN UGR (virtual)

# Ruled real hypersurfaces in $\mathbb CP^n_p$

## Marilena Moruz Al.I. Cuza University of Iasi

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.

Sala EINSTEIN UGR (virtual)

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