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# Generalized Henneberg stable minimal surfaces

In this talk we generalized the classical Henneberg minimal surface by giving an infinite family of complete, finitely branched, non-orientable, stable minimal surfaces in $\mathbb{R}^3$. These surfaces can be grouped into subfamilies depending on a positive integer $m$, which essentially measures the number of branch points. We describe the isometry group of the most symmetric example $H_m$. The surfaces $H_m$ can also be seeing either as the unique solution to a Björling problem for an hypocycloid of $m+1$ cups if $m$ is even or as the conjugate minimal surface to the unique solution to a Björling problem for an hypocycloid of $2m+2$ cups if $m$ is odd.

# Conformal structures with an infinitesimal symmetry

## Omid MakhmaliInstitute of Mathematics of the Polish Academy of Science

We interpret the property of having an infinitesimal symmetry as a variational property in certain geometric structures. This is achieved by establishing a one-to-one correspondence between a class of cone structures with an infinitesimal symmetry and geometric structures arising from certain systems of ODEs that are variational. Such cone structures include conformal pseudo-Riemannian structures and distributions of growth vectors (2,3,5) and (3,6). In this talk we will primarily focus on conformal structures. The correspondence is obtained via symmetry reduction and quasi-contactification. Subsequently, we provide examples of each class of cone structures with more specific properties, such as having a null infinitesimal symmetry, being foliated by null submanifolds, or having reduced holonomy to the appropriate contact parabolic subgroup. As an application, we show that chains in integrable CR structures of hypersurface type are metrizable. This is a joint work with Katja Sagerschnig.

# Lawson's Bipolar Minimal Surfaces in the 5-Sphere

## Melanie RotheTechnische Universität U Darmstadt

Contrary to $\mathbb{R}^n$ as ambient space, there exist compact minimal surfaces in the Euclidean n-sphere $\mathbb{S}^n$. Concerning each topological class, the variety of such examples is very limited, especially in the case of higher codimension. Regarding the latter, H. B. Lawson has shown in 1970 that every minimal immersion $\psi:\Sigma\to\mathbb{S}^3$ of a 2-manifold $\Sigma$ induces a minimal immersion $\widetilde{\psi}:\Sigma\to\mathbb{S}^5$, describing the associated so-called bipolar surface. Due to the example of the Lawson surfaces $\widetilde{\tau}_{m,k}$, analysed by H. Lapointe in 2008, we know that the topology of the bipolar surface can hereby crucially differ from the surface in the 3-sphere. In this context, we give a topological classification of the bipolar Lawson surfaces $\widetilde{\xi}_{m,k}$ and $\widetilde{\eta}_{m,k}$. Additionally, we provide upper and lower area bounds, and find that these surfaces are not embedded for $m\geq 2$ or $k\geq 2$.

# Melanie Rothe

Despacho: B8 - IMAG

Despacho: IMAG

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