Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets

TL;DR

This paper proves a linear isoperimetric inequality for currents in compact subanalytic sets, extending classical inequalities to singular sets and broad classes of currents, with applications to variational and metric properties.

Contribution

It establishes a linear isoperimetric inequality for currents in compact subanalytic sets, including singular and algebraic sets, which was previously unknown.

Findings

Linear inequality holds for integral, normal, and flat chains in subanalytic sets.

Inequality applies to sets with polynomial singularities, unlike classical inequalities.

Applications to variational and metric properties of subanalytic sets.

Abstract

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.