The neck transfer of a plumbing node on the circle Sobolev spaces #
The neck of the plumbing xy = a, 0 < |a| < 1, is the annulus |a| ≤ |x| ≤ 1 in the first
branch coordinate, the second branch coordinate being y = a / x. Its two outer boundary circles
are |x| = 1 and |y| = 1. A function G(y) = ∑ₙ Gₙ yⁿ of the second coordinate, read on the
circle |x| = 1, becomes x ↦ G(a / x), whose (-n)-th Fourier coefficient is aⁿ Gₙ.
This file studies the resulting operator on the circle Sobolev spaces H^s. The neck transfer
CircleSobolev.neckTransfer a sends u to ∑_{n > 0} aⁿ uₙ e^{-inθ}: it carries the positive
frequencies on one boundary circle to the negative frequencies they produce on the other. For
‖a‖ ≤ 1 it has norm at most ‖a‖, it depends holomorphically on a in the open unit disc, and
its derivative at a = 0 is the operator u ↦ u₁ e^{-iθ}, which is nonzero for E ≠ 0.
The linearized node equation F(x) + G(a / x) = const on the neck reads, in the Fourier
coefficients on the two outer circles, F₋ₙ = -aⁿ Gₙ and G₋ₙ = -aⁿ Fₙ for every n > 0. The
pairs (F, G) satisfying it form CircleSobolev.neckGraph s E a. For ‖a‖ ≤ 1 their negative
frequency parts are -neckTransfer a G and -neckTransfer a F, so the neck graph is the graph,
over the pair of nonnegative Laurent subspaces, of a bounded operator that vanishes at a = 0.
It is a closed complement of the pair of negative Laurent subspaces, and at a = 0 it is the pair
H^s_{≥0} × H^s_{≥0} of boundary values of holomorphic functions on the two branch discs of the
node xy = 0. This is the linear model of the Hardy decomposition at a node.
Main definitions #
TauCeti.CircleSobolev.neckTransfer a: the operatoru ↦ ∑_{n > 0} aⁿ uₙ e^{-inθ}onH^s, for‖a‖ ≤ 1(and0otherwise).TauCeti.CircleSobolev.neckGraph s E a: the pairs(F, G)of elements ofH^swithF₋ₙ = -aⁿ GₙandG₋ₙ = -aⁿ Fₙfor everyn > 0.TauCeti.CircleSobolev.neckPoint a (P, Q): the pair(P - neckTransfer a Q, Q - neckTransfer a P)with nonnegative parts(P, Q); for‖a‖ ≤ 1it lies in the neck graph.
Main statements #
TauCeti.CircleSobolev.norm_neckTransfer_apply_le:‖neckTransfer a u‖ ≤ ‖a‖ ‖u‖.TauCeti.CircleSobolev.toContinuousMap_neckTransfer: fors > 1/2and‖a‖ ≤ 1the continuous function ofneckTransfer a uisx ↦ diskFunction u (a / x) - u₀.TauCeti.CircleSobolev.analyticOnNhd_neckTransfer:a ↦ neckTransfer ais analytic on the open unit disc, as a map to the bounded operators onH^s.TauCeti.CircleSobolev.hasDerivAt_neckTransfer_zero: its derivative at0isu ↦ u₁ e^{-iθ}.TauCeti.CircleSobolev.mem_neckGraph_iff_indicator_Iio_eq: the neck graph is the graph of-neckTransfer aon the nonnegative frequencies, andTauCeti.CircleSobolev.isCompl_neckGraph: it is a complement of the pair of negative Laurent subspaces.TauCeti.CircleSobolev.neckGraph_zero: ata = 0it is the pair of nonnegative Laurent subspaces.TauCeti.CircleSobolev.neckPoint_indicatorProj: for‖a‖ ≤ 1every element of the neck graph is theneckPointof its nonnegative parts, andTauCeti.CircleSobolev.analyticAt_neckPoint:neckPoint a (P, Q)is holomorphic in(a, P, Q)for‖a‖ < 1.TauCeti.CircleSobolev.hasSum_zpow_smul_coeff_of_mem_neckGraph: fors > 1/2,a ≠ 0and(F, G)in the neck graph, the Laurent seriesG(a / x) = ∑ₙ (a / x)ⁿ Gₙconverges on|x| = 1andF(x) + G(a / x) = F₀ + G₀.
References #
- J. W. Robbin and D. A. Salamon, A construction of the Deligne–Mumford orbifold, J. Eur. Math. Soc. 8 (2006), 611–699, §§9–12 (Hardy decompositions at the nodes of a plumbed Riemann surface).
The neck transfer #
The neck transfer of the plumbing xy = a: the operator on H^s sending u to
∑_{n > 0} aⁿ uₙ e^{-inθ}. For s > 1/2 and complete E, on a nonnegative-frequency u, the
boundary value of a holomorphic function U on the unit disc, it gives the values of
x ↦ U(a / x) - U(0) on the unit circle (CircleSobolev.toContinuousMap_neckTransfer). It is
defined for ‖a‖ ≤ 1, where its norm is at most ‖a‖ (CircleSobolev.opNorm_neckTransfer_le);
for ‖a‖ > 1 it is the junk value 0.
Instances For
The Fourier coefficients of the neck transfer: the (-n)-th one, n > 0, is aⁿ uₙ, and the
nonnegative ones vanish.
The neck transfer has no nonnegative frequencies.
The neck transfer moves the n-th Fourier coefficient, n > 0, to frequency -n and
multiplies it by aⁿ.
Outside the closed unit disc the neck transfer is the junk value 0.
The neck transfer of a node, a = 0, vanishes.
The neck transfer has norm at most ‖a‖, for every a (with the junk value 0 when
‖a‖ > 1).
The operator norm of the neck transfer is at most ‖a‖.
The neck transfer has only negative frequencies.
The neck transfer only depends on the positive frequencies.
The neck transfer vanishes on elements without positive frequencies.
The neck transfer of a Fourier monomial of positive frequency n is the monomial of frequency
-n scaled by aⁿ.
The neck transfer commutes with the inclusion H^s ⊆ H^t.
Boundary values of the neck transfer. For s > 1/2 and ‖a‖ ≤ 1, the continuous function
on the circle attached to neckTransfer a u is x ↦ diskFunction u (a / x) - u₀, where x is read
as a point fourier 1 x of the unit circle.
Holomorphic dependence on the plumbing parameter #
Holomorphy of the neck transfer. The neck transfer depends holomorphically on the plumbing
parameter a in the open unit disc, as a map to the bounded operators on H^s.
The derivative of the neck transfer at the node a = 0 is the operator u ↦ u₁ e^{-iθ},
which is nonzero for E ≠ 0 although the neck transfer itself vanishes there.
The linearized node equation #
The neck graph of the plumbing xy = a: the pairs (F, G) of elements of H^s whose
Fourier coefficients satisfy F₋ₙ = -aⁿ Gₙ and G₋ₙ = -aⁿ Fₙ for every n > 0. This is the
linearized node equation F(x) + G(a / x) = const on the neck, written in the Fourier
coefficients on its two outer boundary circles
(CircleSobolev.hasSum_zpow_smul_coeff_of_mem_neckGraph).
Instances For
Membership in the neck graph, in terms of the Fourier coefficients.
The neck graph is closed in H^s × H^s.
The neck graph is a graph. For ‖a‖ ≤ 1, a pair (F, G) lies in the neck graph exactly
when the negative-frequency parts of F and G are -neckTransfer a G and -neckTransfer a F;
these only depend on the positive frequencies of G and F.
At the node a = 0 the neck graph is the pair of nonnegative Laurent subspaces, the boundary
values of holomorphic functions on the two branch discs.
Existence in the neck graph. Every pair (F, G) of nonnegative-frequency elements is the
nonnegative part of the element (F - neckTransfer a G, G - neckTransfer a F) of the neck
graph.
Uniqueness in the neck graph. For ‖a‖ ≤ 1, an element of the neck graph is determined by
its nonnegative-frequency parts.
Transversality of the neck graph. For ‖a‖ ≤ 1 the neck graph is a complement of the
pair of negative Laurent subspaces.
The neck graph as a graph over the nonnegative Laurent subspaces #
The pair (P - neckTransfer a Q, Q - neckTransfer a P), whose nonnegative parts are (P, Q)
(CircleSobolev.indicatorProj_sub_neckTransfer). For ‖a‖ ≤ 1 it lies in the neck graph
(CircleSobolev.neckPoint_mem_neckGraph).
Instances For
The nonnegative part of P - neckTransfer a u is P.
For ‖a‖ ≤ 1, every element of the neck graph is the neckPoint of its nonnegative parts.
neckPoint a (P, Q) depends holomorphically on (a, P, Q) for a in the open unit disc.
Boundary values in the neck graph are differences of the two branch disc functions.
This formula also makes sense at a = 0 and uses no Laurent evaluation at a zero coordinate.
The linearized node equation. For s > 1/2, a ≠ 0 and (F, G) in the neck graph,
the Laurent series G(a / x) = ∑ₙ (a / x)ⁿ Gₙ converges at every point x = fourier 1 x of the
unit circle, and F(x) + G(a / x) = F₀ + G₀, where F(x) is the continuous function attached
to F.