Event Details

Seminario de Geometría

Speaker: Melanie Rothe.

Abstract: 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
\(k\geq 2\).