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Doctoral Thesis
Full name
Julio Cesar Correa Hoyos
Knowledge Area
Date of Defense
São Paulo, 2020
Nardulli, Stefano (President)
Guimarães, Maria Fernanda Elbert
Pimentel, Edgard Almeida
Santos, Walcy
Terra, Glaucio
Title in English
Intrinsic geometry of varifolds in Riemannian manifolds: monotonicity and Poincare-Sobolev inequalities
Keywords in English
Analysis on manifolds
Calculus of variations
First variation of a varifold
Geometric measure theory
Metric geometry
Michael-Simon inequality
Poincare and Sobolev-type inequalities
Abstract in English
We prove a Poincare, and a general Sobolev type inequalities for functions with compact support defined on a $k$-rectifiable varifold $V$ defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature bounded above. Our techniques allow us to consider Riemannian manifolds $(M^n,g)$ with $g$ of class $C^2$ or more regular, avoiding the use of Nash's isometric embedding theorem. Our analysis permits to do some quite important fragments of geometric measure theory also for those Riemannian manifolds carrying a $C^2$ metric $g$, that is not $C^{k+\alpha}$ with $k+\alpha>2$. The class of varifolds we consider are those which first variation $\delta V$ lies in an appropriate Lebesgue space $L^p$ with respect to its weight measure $\|V\|$ with the exponent $p\in\R$ satisfying $p>k$.
Title in Portuguese
Geometria intrínsica de varifolds em variedades Riemannianas: monotonia e desigualdades do tipo Poincaré-Sobolev
Keywords in Portuguese
Análise em variedades
Cálculo das variações
Desigualdade de Michael-Simon
Desigualdades do tipo Poincaré-Sobolev
Geometria métrica
Primeira variação de uma varifold
Teoria geométrica da medida
Abstract in Portuguese
São provadas desigualdades do tipo Poincaré e Sobolev para funções com suporte compacto definidas em uma varifold $k$-rectificavel $V$ definida em uma variedade Riemanniana com raio de injetividade positivo e curvatura secional limitada por cima. As técnicas usadas permitem considerar variedades Riemannianas $(M^n,g)$ com métrica $g$ de classe $C^2$ ou mais regular, evitando o uso do mergulho isométrico de Nash. Dita análise permite refazer alguns fragmentos importantes da teoria geométrica da medida também no caso de variedades Riemannianas que admitem uma métrica $C^2$, que possivelmente não é $C^{k+\alpha}$, com $k+\alpha>2$. A classe de varifolds consideradas, são aquelas em que sua primeira variação $\delta V$ está em um espaço de Labesgue $L^p$ com respeito à sua medida de massa $\|V\|$ com expoente $p\in\R$ satisfazendo $p>k$.
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