25.11.3 Curves and zero-volume limits9AB6A1306853

The inverse maximum principle in the plane, lemma 5.27.14, is the two-dimensional case of the result stated by Brezis and Ponce [Theorem 3.1, BP04], who attribute it to Dupaigne and Ponce; the proof given here uses harmonic barriers on shrinking neighborhoods of a compact polar set. Applied to differences of potentials, it shows that the polar curvature measures are ordered as the potentials are, and subtracting a common Green potential gives the formula (5.27.16) for the maximum. Since these local facts are general, they are proved in section 5.21.

The classification of models on curves in section 13.28 combines this order of polar measures with the Green representation of subsection 20.3.9. Every prescribed polar measure gives an ordinary model equivalent to its Green potential, and the characterization of envelopes of positive mass, together with uniqueness of harmonic functions at zero mass, identifies every ordinary envelope with such a model. The formula for the maximum then turns the upper-volume comparison into the total variation of measures, and completeness, geodesics and the nearest ideal model follow from the corresponding operations on measures. The exact defect of analytic approximation is the nonatomic polar mass, which for the Cantor example is the full curvature mass.

For a finite map of curves, the fiber maximum in 5.43.15 follows from the local branched coordinates and the removable-singularity theorem for subharmonic functions. The weighted trace in 5.43.16 includes the local degrees at atoms; its agreement with current pullback follows from the Riesz representation on a disk and the Poincaré–Lelong formula. These elementary operations give the envelope identities of 18.24.1. Applying the curve classification gives the degree-scaled metric embedding and the averaging retraction of 18.24.2. The component factorization then gives the complete subspace of models descending to the original base in 18.24.3. The cubic example 18.24.4 distinguishes this averaging from the nearest ideal retraction.

The continuity statement for curves, theorem 13.28.8, uses the theorem on convergence in capacity at positive volume in the interior. At zero mass the polar curvature is the whole current, which gives convergence of the currents in total variation. The local estimate lemma 5.27.11, which follows from the Green representation on a disk, the mean-value estimate for harmonic functions and the weighted Chern–Levine–Nirenberg inequality, gives convergence in capacity at this boundary. Moving a pole distinguishes this topology from that of total variation, while the approximation by finitely many atoms in theorem 18.6.4 distinguishes weak convergence of the curvature currents from convergence in capacity.

The correspondences between backgrounds in section 13.29 follow from the same classification by polar measures, which smooth gauges extend to every smooth background of positive area on a curve. Changes of background with equal area preserve total variation, whereas rescaling the area gives homotheties compatible with extraction of atoms and hence with ideal projection. These correspondences are realized by a smooth gauge, obtained from the scalar Poisson equation, followed by ordinary projection. Comparing the new obstacle with its maximum and minimum gives a uniform bound, which yields continuity in $L^1$ and in capacity up to the boundary of zero mass. Rationality is needed only for the assertion that analytic singularities are preserved under rescaling of the area, because analytic singularities in this book have rational coefficients. At zero area, positivity and Stokes' theorem leave a single normalized potential. The general convergence statement theorem 5.29.23, proved in section 5.21, follows from compactness of normalized potentials, uniqueness of harmonic functions and the local estimate relating total variation and capacity. It also gives the collapse of the model representatives as the unscaled metric diameters tend to zero.