Intrinsic geometry of varifolds in Riemannian manifolds: monotonicity and Poincare-Sobolev inequalities

• Autor(a) principal: Hoyos, Julio Cesar Correa
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-14082020-141207/
• topic: Análise em variedades; Analysis on manifolds; Cálculo das variações; Calculus of variations; Desigualdade de Michael-Simon; Desigualdades do tipo Poincaré-Sobolev; First variation of a varifold; Geometria métrica; Geometric measure theory; Metric geometry; Michael-Simon inequality; Poincare and Sobolev-type inequalities; Primeira variação de uma varifold; Teoria geométrica da medida

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 $p>k$.