Please use this identifier to cite or link to this item: http://hdl.handle.net/1843/51002
Type: Tese
Title: Topological obstructions to the existence of metrics with non-negative or positive scalar curvature and mean convex boundary
Authors: Franciele Conrado dos Santos
First Advisor: Ezequiel Rodrigues Barbosa
First Referee: Marcos da Silva Montenegro
Second Referee: Almir Rogério Silva Santos
Third Referee: Heleno da Silva Cunha
metadata.dc.contributor.referee4: Ivaldo Paz Nunes
metadata.dc.contributor.referee5: Maria de Andrade Costa e Silva
Abstract: In this work we study the geometry of compact and orientable $n$-dimensional manifolds with non-empty boundary $(M,\partial M)$ such that there is a non-zero degree map $F:(M,\partial M)\rightarrow (\Sigma\times T^{n-2},\partial\Sigma\times T^{n-2})$, where $(\Sigma,\partial \Sigma)$ is a compact, connected and orientable surface with non-empty boundary and $3\leq n\leq 7$. We show that depending on the topology of $\Sigma$, the existence of this non-zero degree map $F$ is a topological obstruction to the existence of a metric in $M$ with positive or non-negative scalar curvature and mean convex boundary. More precisely, we show that \begin{enumerate} \item If $\Sigma$ is neither a disk nor a cylinder then $M$ does not admit a metric with non-negative scalar curvature and mean convex boundary. \item If $\Sigma$ is not a disk then $M$ does not admit a metric with positive scalar curvature and mean convex boundary. Furthermore, every metric in $M$ with non-negative scalar curvature and mean convex boundary is Ricci-flat with totally geodesic boundary. \end{enumerate} Finally, we study the case in which $\Sigma$ is a disk. In this case we consider a metric $g$ in $M$ with positive scalar curvature and mean convex boundary (i.e., $R_g^M>0$ and $H_g^{\partial M}\geq 0$) and we define $\mathcal{F}_M$ be the set of all immersed disks in $M$ whose boundaries are curves in $\partial M$ that are homotopically non-trivial in $\partial M$. We show that \begin{equation}\label{aa} \frac{1}{2}\inf R_g^M \mathcal{A}(M,g)+\inf H_g^{\partial M}\mathcal{L}(M,g)\leq 2\pi \end{equation} \noindent where \[\mathcal{A}(M,g)=\inf_{\Sigma\in \mathcal{F}_M} |\Sigma|_g \ \ \text{e} \ \ \mathcal{L}(M,g)=\inf_{\Sigma\in \mathcal{F}_M} |\partial \Sigma|_g.\] Moreover, if the boundary $\partial M$ is totally geodesic and the equality holds in $(\ref{aa})$, then universal covering of $(M,g)$ is isometric to $(\mathbb{R}^n\times \Sigma_0, \delta+g_0)$, where $\delta$ is the standard metric in $\mathbb{R}^n$ and $(\Sigma_0,g_0)$ is a disk with constant Gaussian curvature $\frac{1}{2}\inf R^M_g$ and $\partial\Sigma_0$ has null geodesic curvature in $(\Sigma_0,g_0)$.
Abstract: Neste trabalho vamos estudar a geometria de variedades n-dimensional orient´aveis e compactas com bordo n˜ao-vazio (M, ∂M) tais que existe uma aplica¸c˜ao de grau diferente de zero F : (M, ∂M) → (Σ×T n−2 , ∂Σ×T n−2 ), onde (Σ, ∂Σ) ´e uma superf´ıcie compacta, conexa, orient´avel com bordo n˜ao-vazio e 3 ≤ n ≤ 7. Mostramos que dependendo da topologia de Σ, a existˆencia desta aplica¸c˜ao de grau diferente de zero F ´e uma obstru¸c˜ao topol´ogica para existˆencia de uma m´etrica em M com curvatura escalar positiva ou n˜ao-negativa e bordo mean convexo. Mais precisamente, mostramos que 1. Se Σ n˜ao ´e um disco e nem um cilindro ent˜ao M n˜ao admite uma m´etrica com curvatura escalar n˜ao-negativa e bordo mean convexo. 2. Se Σ n˜ao ´e um disco ent˜ao M n˜ao admite uma m´etrica com curvatura escalar positiva e bordo mean convexo. Al´em disso, toda m´etrica em M com curvatura escalar n˜aonegativa e bordo mean convexo ´e Ricci-flat com bordo totalmente geod´esico.. Por fim, estudamos o caso em que Σ ´e um disco. Neste caso consideramos uma m´etrica g em M com curvatura escalar positiva e bordo mean convexo(isto ´e, RM g > 0 e H∂M g ≥ 0) e definimos FM como sendo o conjunto de todos os discos imersos em M cujos bordos em ∂M s˜ao homotopicamente n˜ao-triviais em ∂M. Mostramos que 1 2 inf R M g A(M, g) + inf H ∂M g L(M, g) ≤ 2π (2) onde A(M, g) = inf Σ∈FM |Σ|g e L(M, g) = inf Σ∈FM |∂Σ|g. Al´em disso, se ∂M ´e totalmente geod´esico e vale a igualdade em (2), ent˜ao o recobrimento universal de (M, g) ´e isom´etrico a (R n×Σ0, δ+g0), onde δ ´e a m´etrica canˆonica de R n e (Σ0, g0) v vi ´e um disco com curvatura Gaussiana constante 1 2 inf RM g e ∂Σ0 tem curvatura geod´esica nula em (Σ0, g0).
Subject: Matemática - Teses
Espaços de curvatura constante - Teses
Variedades topológicas. - Teses.
language: eng
metadata.dc.publisher.country: Brasil
Publisher: Universidade Federal de Minas Gerais
Publisher Initials: UFMG
metadata.dc.publisher.department: ICX - DEPARTAMENTO DE MATEMÁTICA
metadata.dc.publisher.program: Programa de Pós-Graduação em Matemática
Rights: Acesso Aberto
URI: http://hdl.handle.net/1843/51002
Issue Date: 31-Jan-2010
Appears in Collections:Teses de Doutorado

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