lemmaroutine

Fiber Hessian geometry and Jacobian

Shows rr is uniformly convex on fibers with Jacobian ≍a2s2d\asymp a^2s^{2d}
gives Hessian and determinant sizes for energy and barrier
Lemma·lem:fiber-geometry·§4.1 The geometry of the fibers

Statement (paper)

The spatial Hessian Hfib(r)H_{\rm fib}(r) is positive definite. The full H(r)H(r) is positive semidefinite with a one-dimensional quaternionic kernel contained in both G(r)⊥G(r)^\perp and ker⁡H(∣Φ∣2)\ker H(|\Phi|^2). On the closed fibers, uniformly for a<s<4a<s<4, Hfib(r)≥cs−2I,Gfib(r)Gfib(r)∗≤Hfib(r), dP  dZ≍a2s2d dW  dQ,det⁡Hfib(r)≍a−2s−2d.\begin{aligned} H_{\rm fib}(r)&\ge c s^{-2}I,\qquad G_{\rm fib}(r)G_{\rm fib}(r)^*\le H_{\rm fib}(r),\\ \dd P\,\dd Z&\asymp a^2s^{2d}\dd W\,\dd Q,\qquad \det H_{\rm fib}(r)\asymp a^{-2}s^{-2d}. \end{aligned} The spatial Hfib(∣Φ∣2)H_{\rm fib}(|\Phi|^2) has rank one with nonzero eigenvalue comparable to one. Also ∣G(∣W∣2)∣≲s−1|G(|W|^2)|\lesssim s^{-1}.

In plain English

The fiber Hessian Hfib(r)H_{\rm fib}(r) is positive definite, while the full H(r)H(r) is positive semidefinite with a one-dimensional quaternionic kernel lying in G(r)⊥G(r)^\perp and ker⁡H(∣Φ∣2)\ker H(|\Phi|^2). On closed fibers with a<s<4a<s<4, Hfib(r)≥cs−2IH_{\rm fib}(r)\ge cs^{-2}I, Gfib(r)Gfib(r)∗≤Hfib(r)G_{\rm fib}(r)G_{\rm fib}(r)^*\le H_{\rm fib}(r), the Jacobian satisfies  dP  dZ≍a2s2d dW  dQ\dd P\,\dd Z\asymp a^2s^{2d}\dd W\,\dd Q and det⁡Hfib(r)≍a−2s−2d\det H_{\rm fib}(r)\asymp a^{-2}s^{-2d}. Also Hfib(∣Φ∣2)H_{\rm fib}(|\Phi|^2) has rank one with nonzero eigenvalue comparable to one, and ∣G(∣W∣2)∣≲s−1|G(|W|^2)|\lesssim s^{-1}.

Why it's here

This lemma collects the quantitative fiber geometry that the later energy and barrier arguments rely on: uniform convexity of rr on fibers, control of its gradient by its Hessian, and the size of the volume element and determinant. These follow from the Hessian calculus (positivity, product rule, the holomorphic-square formula H(∑∣Fν∣2)=4J∗JH(\sum|F_\nu|^2)=4J^*J) applied to the explicit coordinates. They supply the Hessian lower bounds and determinant comparisons used in the energy-balance proposition and the fixed-barrier estimates.
Classified routine:
Direct computation in the paper's explicit fiber coordinates using the earlier Hessian calculus lemma (positivity, product rule, and the holomorphic-square formula H(∑∣Fν∣2)=4J∗JH(\sum|F_\nu|^2)=4J^*J); it is quantitative bookkeeping rather than a new idea.

Uses

lemHessian calculus in quaternionic coordinates

Used by

propEnergy flux identity with full wall normalpropEnergy barrier on a fixed fiberclaimLaplacian of segment energy equals full determinantclaimCurrent and density bounds with wall decayclaimHessian lower bound for rP_*claimEnergy and wall flux boundsclaimInterior energy lower bound via ball quadratic

Sources

PaperOpen
Section
§4.1 The geometry of the fibers
Page
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Label
lem:fiber-geometry
Line
sections/energy.tex:49
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