A sessão de Geometria Diferencial ocorrerá nos dias 21 e 22 de Janeiro no Instituto de Matemática da Universidade Federal de Alagoas e será um evento satélite do X Workshop on Differential Geometry, que será realizado no Hotel Ponta Verde Maceió, de 10 a 14 de março de 2025.

Geometria Diferencial

Franciele Conrado (UFS)

TÍTULO: The Wasserstein distance for Ricci shrinkers

RESUMO: Let $(M^n,g,f)$ be a Ricci shrinker such that $\textrm{Ric}_f=\frac{1}{2}g$ and the measure induced by the weighted volume element $(4\pi)^{-\frac{n}{2}}e^{-f}dv_{g}$ is a probability measure. Given a point $p\in M$, we consider two probability measures defined in the tangent space $T_pM$, namely the Gaussian measure $\gamma$ and the measure $\overline{\nu}$ induced by the exponential map of $M$ to $p$. In this talk we will prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric between the measures $\overline{\nu}$ and $\gamma$, and which also elucidates the rigidity implications resulting from this estimate. This is a joint work with Detang Zhou.

Allan George (UFPB)

TÍTULO: On Rigidity of Sub-Static Systems

RESUMO: Sub-static manifolds can be viewed as special solutions to the Einstein equations in the presence of stress-momentum fields satisfying a certain dominant energy condition. In this talk, we address the rigidity problem for such manifolds, possibly with a non-empty boundary. First, we establish certain integral inequalities that extend the works of Chrúsciel and Boucher-Gibbons-Horowitz to sub-static manifolds. Even in the static vacuum case, the obtained inequalities improve on the previously known ones. Next, we present local and global splitting theorems under the assumption of the existence of compact minimal hypersurfaces. Finally, we analyze the system arising from static solutions to the Einstein field equations coupled with a σ-model. This is joint work with G. Colombo (Università di Napoli, Italy), L. Mari (Università di Milano, Italy), and M. Rigoli (Università di Milano, Italy).

Eddygledson Gama (UFPE)

TÍTULO: Uniqueness of tangent planes at infinite time for collapsed self-translating solitons

RESUMO: The main goal of this talk is to prove the uniqueness of the asymptotic planes of complete translating solitons with finite genus, width, and entropy. If time allows, we will also provide some applications of this uniqueness result. This is joint work with Francisco Martin and Niels M. Møller.

Flávio França (URCA)

TÍTULO: An extension of Liebmann’s Theorem

RESUMO: 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 work, we introduce an extension of the Liebmann’s Theorem, focusing on surfaces with boundaries. Specifically, we demonstrate that a locally convex, embedded compact connected CMC surface, bounded by a convex curve, lives in a half space of R^3. In particular, we conclude that spherical caps are the only locally convex, embedded compact connected nonzero CMC surface bounded by a circle.

Wagner Costa-Filho (UFAL-Arapiraca)

TÍTULO: Uma observação sobre variedades m-quasi-Einstein

RESUMO: Consideremos uma variedade Riemanniana M fechada m-quasi-Einstein, com campo vetorial associado X. Mostraremos que a curvatura escalar de M é constante se, e somente se, X é Killing.

Matheus Martins (UFAL)

TÍTULO: Index estimates of compact hypersurfaces in smooth metric measure spaces

RESUMO: In this talk, we investigate the spectra of the stability and Hodge–Laplacian operators on a compact manifold immersed as a hypersurface in a smooth metric measure space, possibly with singularities. Using ideas developed by A. Ros and A. Savo, along with an ingenious computation, we have obtained a comparison between the spectra of these operators. As a byproduct of this technique, we have deduced an estimate of the Morse index of such hypersurfaces.

Marcio Silva Santos (UFPB)

TÍTULO: Rigidez de variedades quasi-Einstein

RESUMO: Nesta palestra abordaremos resultados de classificação para variedades m-quasi Einstein. As nossas principais ferramentas são fórmulas integrais, provenientes da Fórmula da Divergência, que permitem entender a geometria dessa classe de variedades a partir da geometria de sua fronteira.

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