Manuscript XVI-J@1 · XVI-J.tex · lemma · 6c1f306d

Lemma: Closed Hessian formula

Independent polynomial proofs of the numerical tables › Closed formula for the Hessian · lines 810–872 · proof included or linked

XVI-J@1#lem:closedL, XVI-J@1#eq:Lclosed, XVI-J@1#eq:F

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For $N_{A} = η A$ , $N_{C} = η C$ ,

L_{h_{*}} (A, C) = tr N_{A} tr N_{C} - 2 tr (N_{A} N_{C}) - 4 (N_{A} N_{C} + N_{C} N_{A})^{1}_{1} .

Equivalently, if $H = \sum_{i} z_{i} B_{i}$ , its quadratic polynomial is

F (H) = - 4 ((η H)^{2})^{1}_{1} - tr ((η H)^{2}) + \frac{1}{2} (tr (η H))^{2}, L_{ij} = \partial_{z_{i}} \partial_{z_{j}} F .

Proof. Since only the residual second derivative contributes, evaluate at $w^{1} = 0$ , where the residual field and its first derivatives vanish. For

g_{λ} = η + λ (h_{*} + H),

one has there

g_{λ} = η + λ H, \partial g_{λ} = 0, \partial_{1}^{2} g_{λ} = 2 λ η .

The Ricci formula (eq:Ricci) gives, with $U = (η + λ H)^{- 1}$ ,

R (g_{λ}) = 2 λ ((U η U)^{11} - U^{11} tr (U η)) .

Write $N = η H$ , $t = tr N$ , $v = tr (N^{2})$ , and $m = (N^{2})^{1}_{1}$ . The expression in parentheses is

3 + λ (- 2 N^{1}_{1} - t) + λ^{2} (m + t N^{1}_{1} + v) + O (λ^{3}) .

Multiplication by

- det (η + λ H) = 1 + \frac{λ t}{2} + λ^{2} (\frac{t ^{2}}{8} - \frac{v}{4}) + O (λ^{3})

leaves second-order coefficient $m + v /4 - t^{2} /8$ . The signed cubic density is therefore the quadratic expression $- 4 m - v + t^{2} /2$ at this residual jet. Differentiating it twice proves both formulas.

Displayed formulas in this passage

XVI-J@1#eq:Lclosed · line 813
$L_{h_{*}} (A, C) = tr N_{A} tr N_{C} - 2 tr (N_{A} N_{C}) - 4 (N_{A} N_{C} + N_{C} N_{A})^{1}_{1} .$
XVI-J@1#eq:F · line 822
$F (H) = - 4 ((η H)^{2})^{1}_{1} - tr ((η H)^{2}) + \frac{1}{2} (tr (η H))^{2}, L_{ij} = \partial_{z_{i}} \partial_{z_{j}} F .$
unlabeled · line 837
$g_{λ} = η + λ (h_{*} + H),$
unlabeled · line 841
$g_{λ} = η + λ H, \partial g_{λ} = 0, \partial_{1}^{2} g_{λ} = 2 λ η .$
unlabeled · line 848
$R (g_{λ}) = 2 λ ((U η U)^{11} - U^{11} tr (U η)) .$
unlabeled · line 856
$3 + λ (- 2 N^{1}_{1} - t) + λ^{2} (m + t N^{1}_{1} + v) + O (λ^{3}) .$
unlabeled · line 861
$- det (η + λ H) = 1 + \frac{λ t}{2} + λ^{2} (\frac{t ^{2}}{8} - \frac{v}{4}) + O (λ^{3})$

Lemma: Closed Hessian formula

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