lemmaroutine

Hessian calculus in quaternionic coordinates

Gives Hessian rules: Hermitian, trace

\Delta u

, product rule,

4J^*J

calculus toolkit for fibers, energy and Dirichlet arguments

Lemma·lem:hessian-calculus·§3.1 The Hessian in real and complex coordinates

Statement (paper)

In either number of slots,

H(u)

is Hermitian and

\tr H(u)=\Delta u

. The linear map from the real Hessian to

H(u)

sends

\xi\xi^{\mathsf t}

gg^*

, where

g_a=\sum_\alpha e_\alpha\xi_{a\alpha}

. It sends nonnegative real forms to nonnegative quaternionic forms. The product rule is

H(uv)=uH(v)+vH(u)+G(u)G(v)^*+G(v)G(u)^* .

In four slots

H

is covariant under quaternionic unitary changes of coordinates, and in three slots under real orthogonal changes of the coordinate index applied in all slots. If

u

depends only on

P\in\CC_{\mathbf k}^m

, then

G_a(u)=2(\partial_{\bar P_a}u)\mathbf i,\qquad H_{ab}(u)=4\partial_{\bar P_a}\partial_{P_b}u .

In particular, a holomorphic map

F=(F_1,\ldots,F_l)

with complex Jacobian

J_{\nu a}=\partial_{P_a}F_\nu

satisfies

H\Big(\sum_\nu|F_\nu|^2\Big)=4J^*J .

For every smooth real function

\rho

F

G(\rho\circ F)

belongs to the quaternionic image of

J^*

In plain English

The quaternionic (or three-slot) Hessian

H(u)

is Hermitian with trace

\Delta u

, maps real nonnegative Hessians to nonnegative forms, obeys the product rule

H(uv)=uH(v)+vH(u)+G(u)G(v)^*+G(v)G(u)^*

, and is covariant under the natural unitary/orthogonal coordinate changes. For functions of a complex variable

P

H_{ab}(u)=4\partial_{\bar P_a}\partial_{P_b}u

, so

H(\sum|F_\nu|^2)=4J^*J

for holomorphic

F

, and

G(\rho\circ F)

lies in the quaternionic image of

J^*

Why it's here

This lemma collects the basic calculus rules for

H

and

G

that the later arguments use freely: positivity, the product rule, coordinate invariance, and the complex-coordinate formulas. The identity

H(\sum|F_\nu|^2)=4J^*J

and the image statement for

G(\rho\circ F)

give the Hessian structure for composite and holomorphic-square functions, which feeds the fiber geometry, energy balance, barrier and Dirichlet-problem arguments.}

Classified routine:

Basic calculus identities (Hermitian symmetry, trace equals Laplacian, product rule, covariance, complex-coordinate formulas) for the quaternionic Hessian, verified by direct computation in the style of the classical complex Hessian/Levi form calculus; this is bookkeeping rather than a new idea.

Uses

Nothing else in the paper

Used by

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Sources

PaperOpen

Section: §3.1 The Hessian in real and complex coordinates
Page: —
Label: lem:hessian-calculus
Line: sections/calculus.tex:36

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