Lecture summaries

Abstract. In this we present a study of k-dimensional stationary (i.e. of zero mean curvature vector field) spacelike submanifolds of the Rotational Subspacetimes of L^{n+2}. which can also be regarded as spherical RW spacetimes. This way we prove a wide extension of Markvorsen Theorem giving a characterization of stationary spacelike submanifolds of spherical RW spacetimes.

Abstract. A soliton is a special solution to a partial differential equation that maintains its shape and moves at a constant velocity. In the context of mean curvature flow, a translating soliton is a solution to the mean curvature flow equation that moves by a constant velocity in the direction of a vector. Translating solitons are particularly interesting because they provide insights into the behavior of evolving surfaces. On the other hand, we say that a surface is semi-graphical if when we remove a discrete set of vertical lines, then the resulting surface is the graph of a smooth function. We are going to provide a classification of all the semi-graphical translator in Euclidean 3-space. First, we will describe a comprehensive zoo of all examples of this type of translators, and then we will focus on classification arguments. We will conclude with some open problems. This talk summarizes various joint works with D. Hoffman and B. White, on one hand, and with M. Saez and R. Tsiamis, on the other.

Abstract. Our purpose is to investigate the geometry of complete spacelike hypersurfaces immersed in the anti-de Sitter space. In this setting, we start by proving rigidity results for such hypersurfaces, under suitable constraints on their higher-order mean curvatures. Afterward, we obtain a lower estimate for the index of minimum relative nullity for r-maximal spacelike hypersurfaces and a nonexistence result for 1-maximal spacelike hypersurfaces of the anti-de Sitter space  Finally, we employ a technique due to Aledo and Alías to prove some curvature estimates for complete spacelike hypersurface of the anti-de Sitter space and, as consequence, we get further nonexistence results concerning these spacelike hypersurfaces. In particular, we show the nonexistence of complete maximal spacelike hypersurfaces in certain open regions of the anti-de Sitter space. Our approach is mainly based on a suitable extension of the generalized maximum principle of Omori and Yau due to Alías, Impera, and Rigoli.

Abstract. In this talk, we recall the geometric flow associated with the high-order mean curvature and present some interesting results in the literature. Next, we present some new rigidity results under suitable hypotheses.

Abstract. In this talk, we examine the behavior of conformal metrics $g^{\mu, \lambda}$ with prescribed sign-changing functions of Gaussian curvature $f + \mu$ and geodesic curvature $h + \lambda$ on surfaces with possibly many boundary components and higher genus. Utilizing a refined mountain-pass technique, we obtain additional estimates for the corresponding saddle-type ("large") solutions $u_{\mu,\lambda}$ that allow us to characterize their "bubbling behavior."

Abstract. This lecture aims to present some results about the classification of Einstein hypersurfaces of the warped product, where the fiber is a Riemannian space form. In particular, we show that such hypersurface has at most three distinct principal curvatures and is locally a multiply warped product with at most two fibers. We also show that exactly one or two principal curvatures on an open set imply constant sectional curvature on that set. For exactly three distinct principal curvatures this is no longer true, and we classify such hypersurfaces provided it does not have constant sectional curvature and a certain principal curvature vanishes identically.

Abstract. In this lecture, we will present an integral inequality for closed linear Weingarten m-submanifolds with parallel normalized mean curvature vector field (pnmc lw-submanifolds) in the product spaces M^n(c)×R, n > m ≥ 4, where M^n(c) is a space form of constant sectional curvature c ∈ {1, 1}. As an application, it is shown that the sharpness in this inequality is attained in the totally umbilical hypersurfaces, and a certain family of standard product of spheres when c = 1. In the case where c = −1 , is obtained an integral inequality whose sharpness is attained only in the totally umbilical hypersurfaces.

Abstract. In this talk, we explore the nature of Lens spaces to study the first width of those spaces, more precisely, we use the existence of a sharp sweep out associated to a Clifford torus to provide a simple and pretty application of the Willmore conjecture for the computation of the 1-width of Lens spaces. This is joint work with M. Batista.