claimroutineinferred_dep

Fiber chart and geometric bounds

Builds holomorphic fiber chart B(W)B(W) with volume and Hessian bounds
chart geometry for the current and multipliers
Claim·§E.1 Eigenvectors and the spatial fiber·inside thmLocal smoothing for symmetric pencil waves

Statement (paper)

Let e(W)=(1,W)te(W)=(1,W)^{\mathsf t} and Bj(W)=e(W)tCje(W)/(1+W⋅W)B_j(W)=e(W)^{\mathsf t}C_je(W)/(1+W\cdot W) for W∈CkdW\in\CC_{\mathbf k}^d near zero. Then B(v(ξ))=∇ϕ(ξ)B(v(\xi))=\nabla\phi(\xi) on real eigenvectors, and for a fixed small w0>0w_0>0 the projection of BB onto ω⊥\omega^\perp is a holomorphic coordinate map on a neighborhood of ∣W∣≤w0|W|\le w_0, with [dB(W),ω][dB(W),\omega] and its inverse bounded there. In the chart P=τB(W)+ΦωP=\tau B(W)+\Phi\omega, R2=2τ/(1+W⋅W)R^2=2\tau/(1+W\cdot W), y=∣R∣2≍s=∣τ∣y=|R|^2\asymp s=|\tau|, with a=Kha=Kh, Q=Φ/aQ=\Phi/a and r=∣W∣2/w02+100∣Q∣2−1r=|W|^2/w_0^2+100|Q|^2-1, for a<s<4a<s<4 one has ∣dW∣≲s−1|dW|\lesssim s^{-1}, ∣dΦ∣≲1|d\Phi|\lesssim1 and  dP≍a2s2d dW  dQ\dd P\asymp a^2s^{2d}\dd W\,\dd Q; moreover, with F=(W/w0,10Q)F=(W/w_0,10Q), H(r)=4(dF)∗dFH(r)=4(dF)^*dF has spatial block ≳s−2I\gtrsim s^{-2}I with determinant ≍a−2s−2d\asymp a^{-2}s^{-2d}, satisfies Gfib(r)Gfib(r)∗≤Hfib(r)G_{\rm fib}(r)G_{\rm fib}(r)^*\le H_{\rm fib}(r), the full Hessian has rank nn with quaternionic null line in ker⁡dΦ∩G(r)⊥\ker d\Phi\cap G(r)^\perp, and the spatial Hessian of ∣Φ∣2|\Phi|^2 has rank one with nonzero eigenvalue comparable to one. At W=v(ξ)W=v(\xi) the conormal is dΦ=(ξ⋅dP−ϕ(ξ) dτ)/(ξ⋅ω)d\Phi=(\xi\cdot dP-\phi(\xi)\,d\tau)/(\xi\cdot\omega).

In plain English

Near a top eigenvector, the map B(W)=e(W)tCje(W)/(1+W⋅W)B(W)=e(W)^{\mathsf t}C_je(W)/(1+W\cdot W) gives a holomorphic coordinate chart on the complexified fiber ∣W∣≤w0|W|\le w_0 that restricts to ∇ϕ\nabla\phi on real eigenvectors. In the resulting variables (W,Q)(W,Q) with defining function rr, the volume, differential, and Hessian bounds hold for a<s<4a<s<4, and at W=v(ξ)W=v(\xi) the conormal is dΦ=(ξ⋅dP−ϕ(ξ) dτ)/(ξ⋅ω)d\Phi=(\xi\cdot dP-\phi(\xi)\,d\tau)/(\xi\cdot\omega).

Why it's here

This claim turns the hypotheses on the pencil into the concrete chart geometry that the later steps use. Polarization being a local angular coordinate (the previous claim) shows that the projection of BB onto ω⊥\omega^\perp is a local holomorphic chart. The Hessian calculus then gives H(r)=4(dF)∗dFH(r)=4(dF)^*dF with the stated spatial block, determinant and null line, and also the rank-one Hessian of ∣Φ∣2|\Phi|^2. These are the geometric inputs for the conserved current and multiplier construction in the next claim.
Classified routine:
A computational step inside the pencil-wave proof: it assembles the chart geometry from the preceding claim (polarization as a local angular coordinate) together with the paper's quaternionic Hessian calculus, using the inverse function theorem and direct differential and volume bookkeeping.

Uses

claimPolarization is a local angular coordinatelemHessian calculus in quaternionic coordinates

Used by

claimConserved current with Gaussian multiplier symbols

Sources

PaperOpen
Section
§E.1 Eigenvectors and the spatial fiber
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sections/pencil.tex:80
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

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    outline: "Fiber chart and geometric bounds" rests on claim-thm-pencil-1
  • info
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    outline: "Fiber chart and geometric bounds" rests on lem:hessian-calculus