Program:

Mar 21, 2024

Speaker:  Marcos Paulo Tassi - (Università degli Studi dell’Aquila, UNIVAQ, Italy)

Title: Elliptic Weingarten surfaces in R^3 with convex planar boundary

Abstract: A surface Σ immersed in R^3 is an elliptic Weingarten surface if its principal curvatures k1 and k2 satisfy an equation of the type W(k1, k2) = 0, for some function W : R^2 → R of class C^1 such that (∂W/∂k1)(∂W/∂k2) > 0 on W^{−1}({0}). Known examples of elliptic Weingarten surfaces include minimal and constant mean curvature surfaces, and surfaces of positive constant gaussian curvature. In 1996 A. Ros and H. Rosenberg proved that for a strictly convex curve Γ ⊂ {z = 0} ⊂ R3, there exists a constant h depending only on the curve Γ such that any compact surface embedded in R^3_+ := {z ≥ 0} with constant mean curvature H ≤ h must be topologically a closed disk. In this talk we will present a generalization of Ros-Rosenberg Theorem for elliptic Weingarten surfaces in R^3, discussing its proof, which is based on some geometric analysis techniques as the Maximum Principle and the Alexandrov Reflection Method, and the recent classification of elliptic Weingarten surfaces of revolution obtained by I. Fernandez and P. Mira. This is a joint work with B. Nelli (UNIVAQ) and G. Pipoli (UNIVAQ)

Mar 08, 2024

Speaker:  Marcos Paulo Tassi - (Università degli Studi dell’Aquila, UNIVAQ, Italy)

Title:  Analytic saddle spheres in S^3 are equatorial

Abstract: The Calabi-Almgren Theorem states that any minimal topological sphere immersed in the 3-sphere S3 must be equatorial, i.e., a totally geodesic 2-sphere of S3. In this talk we will present a generalization of the Calabi-Almgren Theorem: any real analytic topological sphere immersed in S3 which is also saddle (i.e., its principal curvatures k1 and k2 satisfy k1k2 ≤ 0 at each point of the surface) must be equatorial. We will also present a sketch of its proof, based on the study of the umbilical points of the saddle sphere and the Poincaré-Hopf Index Theorem and we will discuss smooth, non-equatorial counter-examples. This is a joint work with J.A. Gálvez (UGr) and P. Mira (UPCT).

This seminar, starting from 2023, has been renamed as the "São Carlos Geometry Seminars" and will continue to prioritize research seminars open to students, faculty, and researchers interested in Geometry. Previously, the seminar was known as the "Differential Geometry Seminar," established in 2018 within the scope of the Differential Geometry Research Group at UFSCar in the CNPq Directory and as part of the Thematic Project funded by FAPESP. Since then, under normal circumstances, it took place at the Department of Mathematics at UFSCar. In 2020, during the pandemic, it was conducted virtually. We would like to emphasize that the seminar's objective is to enhance interaction among graduate students, faculty, and researchers in the field of differential geometry through the discussion of common interests. In particular, the lectures do not need to be based on completed works or even authored by the presenter; the idea is for the seminars to stimulate the development of the field. If you would like to receive notifications or participate in the seminars, please email: lobos@ufscar.br."