论文标题
紧凑型亚分析集中正常和积分电流的线性等级不等式
Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets
论文作者
论文摘要
对于光滑紧凑的Riemannian流形$ a $的等量不平等，提供了正$ {\ bf c}（a）$，因此对于任何$ k+1 $ dimentional的积分电流$ s_0 $ in $ a $ in $ a $ in $ a $中的$ s $ in $ in $中的$ s $ in $ \ pottial s = p partial s = \ bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf bf m. {\ bf c}（a）{\ bf m}（\ partial s）^{（k+1）/k} $。尽管这种不平等仍然适用于任何紧凑型Lipschitz社区缩回$ a $，但如果$ a $包含单个多项式奇点，它可能会失败。在这里，将$（k+1）/k $替换为$ 1 $，我们发现线性不等式$ {\ bf m}（s）\ leq {\ bf c}（a）{\ bf m}（\ bf m}（\ partial s）$有效，适用于任何紧凑型algebraic algebraic，sem-emi-algebraic，sem-algebraic，sem-algebraic cop sep sep cup a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a。在这样的集合中，这种线性不平等不仅适用于具有$ \ boldsymbol {z} $系数的积分电流，而且还适用于具有$ \ boldsymbol {r} $系数的正常电流，并且通常适用于在任何完整规范的阿贝尔组中具有系数的普通平坦链。亚分析对$ b \ subset a $的相对版本也是正确的，并且有应用于亚分析集的变异和度量属性。
The isoperimetric inequality for a smooth compact Riemannian manifold $A$ provides a positive ${\bf c}(A)$, so that for any $k+1$ dimensional integral current $S_0$ in $A$ there exists an integral current $ S$ in $A$ with $\partial S=\partial S_0$ and ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)^{(k+1)/k}$. Although such an inequality still holds for any compact Lipschitz neighborhood retract $A$, it may fail in case $A$ contains a single polynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)$ is valid for any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a set, this linear inequality holds not only for integral currents, which have $\boldsymbol{Z}$ coefficients, but also for normal currents having $\boldsymbol{R}$ coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair $B\subset A$ is also true, and there are applications to variational and metric properties of subanalytic sets.
