论文标题
Poincaré和Sobolev类型的不平等现象，用于内在的可整流varifolds
Poincaré and Sobolev type inequalities for intrinsic rectifiable varifolds
论文作者
论文摘要
我们证明了庞加莱和一般的Sobolev类型不等式，用于在$ k $ - 可纠正的varifold $ v $上定义的紧凑型支持的功能，该功能在完整的Riemannian歧管上定义了，并具有正面的注射率半径和截面曲线。我们的技术使我们可以考虑使用$ g $ class $ c^2 $或更多规则的Riemannian歧管$（m^n，g）$，避免使用Nash的等距嵌入定理。我们的分析允许对那些带有$ c^2 $ metric $ g $的Riemannian流形的几何测量理论进行一些非常重要的片段，而不是$ c^{k+α} $，而不是$ k+α> 2 $。我们认为的varifold类是那些首先变体$ΔV$的varifolds，就其重量度量$ \ | v \ | $带有指数$ p \ in \ mathbb {r} $满足$ p> k $。
We prove a Poincaré, 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 $C^{k+α}$ with $k+α>2$. The class of varifolds we consider are those which first variation $δV$ lies in an appropriate Lebesgue space $L^p$ with respect to its weight measure $\|V\|$ with the exponent $p\in\mathbb{R}$ satisfying $p>k$.
