We show that constant mean curvature hypersurfaces in $\mathbb{H}^n\times\mathbb{R}$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces determined by $P$. This is a joint work with B. Nelli and it is the analogous in $\mathbb{H}^n\times\mathbb{R}$ of a result in $\mathbb{R}^3$ by A. Ros and H. Rosenberg.

We construct for every connected surface $S$ of finite negative Euler characteristic and every $H \in [0,1)$, a hyperbolic 3-manifold $N$ of finite volume and a proper, two-sided, totally umbilic embedding $f\colon S\to N$ with mean curvature $H$. Conversely, we prove that a complete, totally umbilic surface with mean curvature $H \in [0,1)$ embedded in a hyperbolic 3-manifold of finite volume must be proper and have finite negative Euler characteristic. Joint work with Colin Adams and William Meeks.

We consider the Gaussian curvature conjecture of a minimal graph \(S\) over the unit disk. First of all we reduce the general conjecture to the estimating the Gaussian curvature of some Scherk's type minimal surfaces over a quadrilateral inscribed in the unit disk containing the origin inside. As an application we improve so far the obtained upper estimates of  Gaussian curvature at the point above the center. Further we obtain an optimal estimate of the Gaussian curvature at the point \(\mathbf{w}\) over the center of the disk, provided \(\mathbf{w}\) satisfies certain ''symmetric'' conditions. The result extends a classical result of Finn and Osserman in 1964. In order to do so, we construct a certain family \(S^t\), \(t\in[t_\circ, \pi/2]\) of Scherk's type minimal graphs over the isosceles trapezoid inscribed in the unit disk. Then we compare the Gaussian curvature of the graph \(S\) with that of \(S^t\) at the point \(\mathbf{w}\) over the center of the disk.

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Translating solution to the mean curvature flow form, together with self-shrinking solutions, the most important class of singularity models of the flow. When a translator arises as a blow-up of a mean convex mean curvature flow, it also naturally satisfies a non-collapsness condition. In this talk, I will report on a recent work with Kyeongsu Choi and Robert Haslhofer, in which we show that every mean convex, non-collapsed, translator in \(\mathbb{R}^4\) is a member of a one parameter family of translators, which was earlier constructed by Hoffman, Ilmanen, Martín and White.

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We shall prove that in the first Heisenberg group with a sub-Finsler structure, a complete, stable, Euclidean Lipschitz and \(H\)-regular surface is a vertical plane. This is joint work with Manuel Ritoré.

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In this talk we study applications of the Kulkarni-Pinkall form to the study of locally strictly convex immersions in \(\mathbb{H}^3\). We deduce a new a priori estimate which in turn allows us to completely solve the asymptotic Plateau problem for \(k\)-surfaces in hyperbolic space as formulated by Labourie. This work has interesting intersections with a paper of Espinar-Gálvez-Mira. This work appears in https://arxiv.org/abs/2104.03181.

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We consider a nilpotent Lie group with a bracket-generating distribution \(\mathcal{H}\) and an asymmetric left-invariant norm \(\|\cdot\|_K\) induced by a convex body \(K\subseteq\mathcal{H}_0\) containing \(0\) in its interior. In this talk, we will associate a left-invariant perimeter functional \(P_K\) to \(K\) following De Giorgi's definition of perimeter and prove the existence of minimizers of \(P_K\) under a volume (Haar measure) constraint. We will also discuss some properties of the isoperimetric regions and the isoperimetric profile.

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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.

Almost all of the extensive research on mean curvature flow has been for surfaces without boundary. However, it is interesting and natural to consider MCF for surfaces with boundary. In this talk, I will describe a useful weak formulation of such flows that gives existence for all time with arbitrary initial data. Furthermore, under rather mild hypotheses on the initial surface, the moving surface remains forever smooth at the boundary, even after singularities may have formed in the interior. On the other hand, if one relaxes those hypotheses, then interesting boundary singularities can occur.

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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We will show that, in contrast with the \(3\)-dimensional case, the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the \(n\)-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for \(n\geq 4\). Moreover, we will show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, \(n\geq 8\), with Morse index equal to \(0\). Also, for \(n\geq 4\), there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. Finally, we will discuss the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.

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In this talk we will construct via Daniel's sister correspondence in \(\mathbb{H}^2\times\mathbb{R}\) a \(2\)-parameter family of Alexandrov-embedded constant mean curvature \(0\,\)<\(\,H\leq 1/2\) surfaces in \(\mathbb{H}^2\times \mathbb{R}\) with \(2\) ends and genus \(0\). They are symmetric with respect to a horizontal slice and \(k\) vertical planes disposed symmetrically. We will discuss the embeddedness of the constant mean curvature surfaces of this family, and we will show that the Krust property does not hold for \(0\,\)<\(\,H\leq 1/2\); i.e, there are minimal graphs over convex domain in \(\widetilde{\text{SL}}_2(\mathbb{R})\) and \(\text {Nil}_3\) whose sister conjugate surface is not a vertical graph in \(\mathbb{H}^2\times\mathbb{R}\).