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. In this talk, we discuss the geometry of compact quasi-Einstein manifolds with boundary. This topic is directly related to warped product Einstein metrics, static spaces, and smooth metric measure spaces. We show that a 3-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature must be isometric to either the standard hemisphere S^3_{+}, or the cylinder IxS^2 with product metric. For dimension n=4, we prove that a 4-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric to either the standard hemisphere S^4_{+}, or the cylinder I x S^3 with product metric, or the product space S^2_{+} x S^2 with the doubly warped product metric. Other related results for arbitrary dimensions are also discussed. This is a joint work with J. Costa and D. Zhou.
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. Abstract: We present existence and classification results for translation solitons (or simply translators) to the mean curvature flow in the hyperbolic 3-space H3. Joint work with R. de Lima and J. P. dos Santos.
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. We show some area estimates for stable CMC hypersurfaces which the ambient manifold has either the scalar or sectional curvature bounded from below. In particular, we focus on immersions in three dimensional Riemannian manifolds. As application we derive upper estimates for the bottom spectrum of these hypersurfaces. This is part of a joint work with M. Ranieri and F. Vitório.
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. Mean curvature flow self-translating solitons are minimal hypersurfaces for a certain incomplete conformal background metric, and are among the possible singularity models for the flow. In the collapsed case, they are confined to slabs in space. The simplest non-trivial such example, the grim reaper curve $\Gamma$ in $\mathbb{R}^2$, has been known since 1956, as an explicit ODE-solution, which also easily gave its uniqueness. We consider here the case of surfaces, where the rigidity result for $\Gamma\times\mathbb{R}$ that we'll show is: The grim reaper cylinder is the unique (up to rigid motions) finite entropy unit speed self-translating surface which has width equal to $\pi$ and is bounded from below. (Joint w/ Impera \& Rimoldi.) Time permitting, we'll also discuss recent uniqueness results in the collapsed simply-connected low entropy case (w/ E.S. Gama \& F. Martín), using Morse theory and nodal set techniques, which extend Chini's classification.
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. In this lecture, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach Serrin's problem in bounded domains of manifolds endowed with a conformal vector field. Our primary tool, in this case, is a new Pohozaev identity, which depends on the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces. This is a joint work with A. Roncoroni (Politecnico di Milano, Italy) and M. Santos (UFPB, Brazil).
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. The Gauss map of constant mean curvature surfaces in a 3-dimensional Lie group equipped with a bi-invariant metric is harmonic, by a Ruh-Vilms type theorem. Thus, it is natural to attempt to relate the second variation of energy with geometric (or topological) invariants. In this talk, we demonstrated that the index of the energy of the Gauss map of these surfaces is bounded from below by a linear function of their genus. This is joint work with M. Cavalcante, W. Costa-Filho, and D. Oliveira.
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. The classical Liebmann’s Theorem asserts that a compact connected convex surface in R^3, with constant mean curvature (CMC), is a totally umbilical sphere. In this presentation, we introduce an extension to Liebmann’s Theorem, focusing on surfaces with boundaries. Specifically, we demonstrate that a locally convex, embedded compact connected CMC surface, bounded by a circle, is necessarily a spherical cap.
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.