Manuscript III@2 · III.tex · section · 82dad3e5

A closed-range membership test

The C1 triangular Cauchy quotient › A closed-range membership test · lines 1890–1910

III@2#eq:range-reconstruction

For arbitrary compact metric data define auxiliary fields

α_{0} α_{i} β_{0} β_{i} = \frac{1}{2} N B_{00}, = N (\partial_{j} A_{ij} - \frac{1}{2} \partial_{i} A_{j j}), = \frac{1}{2} A_{00}, = A_{0 i} - \partial_{i} α_{0} .

Denote this reconstruction by $R (A, B)$ . The entries are smooth and tend to zero at spatial infinity. They may initially be noncompact. The differential expression $g$ can be evaluated on them without admitting them as gauge data.

Displayed formulas in this passage

III@2#eq:range-reconstruction · line 1893
$α_{0} α_{i} β_{0} β_{i} = \frac{1}{2} N B_{00}, = N (\partial_{j} A_{ij} - \frac{1}{2} \partial_{i} A_{j j}), = \frac{1}{2} A_{00}, = A_{0 i} - \partial_{i} α_{0} .$

Nearby in this section

lemma: Lemma: Closed-range test
section: A closed-range membership test
theorem: Theorem: Well-posed reduced Cauchy problem
section: A closed-range membership test
figure: Stagewise Cauchy quotient