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

Thierry De Pauw and Robert M. Hardt

Journal of Singularities
volume 24 (2022), 145-168

Received: 15 February 2022. Received in revised form: 5 September 2022.

DOI: 10.5427/jsing.2022.24f

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Abstract:

The isoperimetric inequality for a smooth compact Riemannian manifold A provides a positive constant c depending only on A, so that whenever a k-dimensional integral current T in A bounds some integral current S in A, S can be chosen to have mass at most c times the (k+1)/k power of the mass of T. 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 this power by one, we verify the linear inequality, the mass of S being bounded by c times the mass of T, is valid for any compact algebraic, semialgebraic, or even subanalytic set A. In such a set, this linear inequality holds not only for integral currents, which have integer coefficients, but also for normal currents having real coefficients and generally for normal flat chains having coefficients in any complete normed Abelian group. A relative version for a subanalytic pair (A,B) is also true, and there are applications to variational and metric properties of subanalytic sets.

2010 Mathematical Subject Classification:

Primary 49Q15, 49Q20, 32B20, 14P10; Secondary 52A40, 49J45

Key words and phrases:

Isoperimetric inequality, normal and integral currents, subanalytic geometry, Plateau problem

Author(s) information:

Thierry De Pauw
Université Paris Cité and
Sorbonne Université CNRS IMJ-PRG
F-75013 Paris, France.
email: depauw@imj-prg.fr

Robert M. Hardt
Department of Mathematics
Rice University
P. O. Box 1892
Houston, Texas 77251
email: hardt@rice.edu