## View abstract

### Session S39 - Differential Equations and Geometric Structures

Tuesday, July 13, 15:00 ~ 15:50 UTC-3

## Curvature lines of compact surfaces of genus $k$.

### Federico Sánchez-Bringas

#### Universidad Nacional Autónoma de México, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakb16e59a6e3cd94e2138ff04fe8fcda1f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb16e59a6e3cd94e2138ff04fe8fcda1f = 's&#97;nch&#101;z' + '&#64;'; addyb16e59a6e3cd94e2138ff04fe8fcda1f = addyb16e59a6e3cd94e2138ff04fe8fcda1f + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_textb16e59a6e3cd94e2138ff04fe8fcda1f = 's&#97;nch&#101;z' + '&#64;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloakb16e59a6e3cd94e2138ff04fe8fcda1f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb16e59a6e3cd94e2138ff04fe8fcda1f + '\'>'+addy_textb16e59a6e3cd94e2138ff04fe8fcda1f+'<\/a>';

We analyze the curvature lines of a compact surface of genus $k \in \mathbb N$ embedded in $\mathbb S^3$ as the link of the real part of the Milnor fibration of a polynomial in $\mathbb C^2$. Since the polynomial is quasi-homogeneous, the link gives rise, under the natural $\mathbb C^*$-action, a family of diffeomorphic surfaces with equivalent curvature lines. We use the symmetries of the model to describe the curvature lines in the cases of genus $k=2,3$. Finally, we study a bifurcation of the case of genus $k=3$ along the umbilic points.

Joint work with Vinicio Gómez Gutiérrez (Universidad Nacional Autónoma de México).