Manuscript VII-A@1 · VII-A.tex · theorem · 11da3fe6

Theorem: Exact first-order plateau repair

Exact first-order repair and the larger-support defect › The plateau theorem · lines 973–1012 · proof included or linked

VII-A@1#thm:first, VII-A@1#eq:firstinvariance, VII-A@1#eq:firstthree

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Let the data satisfy Section sec:data. For every smooth metric coefficient $h$ and every $ξ \in g_{U}$ ,

A_{1}^{(g)} [h + K ξ; ϕ_{0}] = A_{1}^{(g)} [h; ϕ_{0}] .

More explicitly,

g K ξ = K ξ, ϕ_{1, g} [h + K ξ] - ϕ_{1, g} [h] = X_{ξ} ϕ_{0}, w [h + K ξ] - w [h] = ξ .

Consequently, the first-order coefficient of $O_{f} [g]$ is invariant under the infinitesimal transformations admitted by $G_{U}$ .

Proof. All derivatives of $ξ$ are supported in $U$ , where $g = 1$ . Hence $g K ξ = K ξ$ . By linearity and Lemma lem:QK,

ϕ_{1, g} [h + K ξ] - ϕ_{1, g} [h] = - E_{m} Q_{gK ξ}^{(m)} ϕ_{0} = E_{m} P_{m} (X_{ξ} ϕ_{0}) = X_{ξ} ϕ_{0} .

The last equality is valid because $ξ$ vanishes near $Σ_{0}$ . Lemma lem:CK gives the coordinate variation. Substitution into (eq:A1) proves (eq:firstinvariance).

Displayed formulas in this passage

VII-A@1#eq:firstinvariance · line 977
$A_{1}^{(g)} [h + K ξ; ϕ_{0}] = A_{1}^{(g)} [h; ϕ_{0}] .$
VII-A@1#eq:firstthree · line 984
$g K ξ = K ξ, ϕ_{1, g} [h + K ξ] - ϕ_{1, g} [h] = X_{ξ} ϕ_{0}, w [h + K ξ] - w [h] = ξ .$
unlabeled · line 1000
$ϕ_{1, g} [h + K ξ] - ϕ_{1, g} [h] = - E_{m} Q_{gK ξ}^{(m)} ϕ_{0} = E_{m} P_{m} (X_{ξ} ϕ_{0}) = X_{ξ} ϕ_{0} .$

Nearby in this section

section: The plateau theorem