Moments of the Euclidean Yamabe bubbles #
Let U = U_{t,b} be a Yamabe bubble on an n-dimensional real inner product space. This file
computes the radial moments ∫ ‖x - b‖ ^ p … of order p ≥ 0 of its critical density U ^ pₙ,
of its Dirichlet-energy density |dU|², and of its square U². For t > 0,
- the moment of order
pofU ^ pₙis finite whenn ≠ 2andp < n, and scales ast ^ p; - the moment of order
pof|dU|²is finite whenp < n - 2, and also scales ast ^ p; - the moment of order
pofU²is finite whenp < n - 4, and scales ast ^ (p + 2). Thus its second moment has scalet⁴in dimensions at least seven.
The exact values hold for every additive Haar measure and retain the beta integral and the
measure of the unit ball. This form makes both the scale and the loss of integrability at the
critical moment transparent. In particular, the fourth energy moment is covered in dimensions
at least seven. The integrability and the value of ∫ U ^ pₙ in
TauCeti.Analysis.PDE.Yamabe.Bubble.Energy are the moments of order 0.
At the endpoint orders p = n for U ^ pₙ (in dimension n ≠ 2 with n > 0), and p = n - 2
for |dU|² and p = n - 4 for U² (both in dimension n > 2), the moments diverge
logarithmically. For these the file computes the moments truncated to a ball B(b, R) exactly,
in the same dimensions. Each is a constant times t ^ q (log (1 + (R / t)²) - S_j), where q is
n for U ^ pₙ and n - 2 for the other two densities, and S_j is the partial logarithmic
series ∑_{k < j} w ^ (k + 1) / (k + 1) in w = R² / (t² + R²) ∈ [0, 1), with j = n - 1 for
U ^ pₙ and |dU|² and j = n - 3 for U². Hence the truncated moment is the logarithmic term
up to an error of at most the same constant times t ^ q H_j, with H_j the j-th harmonic
number. In dimension six these are the fourth moment of |dU|²
and the second moment of U², which grow like t⁴ log (1 / t) as t → 0.
Main declarations #
TauCeti.Yamabe.integrable_norm_sub_rpow_mul_bubble_rpow_criticalExponentandTauCeti.Yamabe.integral_norm_sub_rpow_mul_bubble_rpow_criticalExponent: moments of the critical density.TauCeti.Yamabe.integrable_norm_sub_rpow_mul_norm_fderiv_bubble_sqandTauCeti.Yamabe.integral_norm_sub_rpow_mul_norm_fderiv_bubble_sq: moments of the energy density.TauCeti.Yamabe.integral_mul_norm_sub_rpow_mul_bubble_sq_eq_mul: scaling of square moments against arbitrary real weights.TauCeti.Yamabe.integrable_norm_sub_rpow_mul_bubble_sqandTauCeti.Yamabe.integral_norm_sub_rpow_mul_bubble_sq: the finite moments ofU².TauCeti.Yamabe.tendsto_integral_norm_sub_rpow_mul_bubble_rpow_criticalExponentandTauCeti.Yamabe.tendsto_integral_norm_sub_rpow_mul_norm_fderiv_bubble_sq: every integrable positive-order moment tends to zero with the bubble scale.TauCeti.Yamabe.setIntegral_ball_norm_sub_rpow_mul_bubble_rpow_criticalExponent,TauCeti.Yamabe.setIntegral_ball_norm_sub_rpow_mul_norm_fderiv_bubble_sqandTauCeti.Yamabe.setIntegral_ball_norm_sub_rpow_mul_bubble_sq, with theirabs_…_sub_mul_log_lecompanions: the truncated logarithmic moments ofU ^ pₙ,|dU|²andU². They specialize the truncated moments of the powers of the bubble profile,TauCeti.setIntegral_ball_norm_sub_rpow_mul_div_add_norm_sub_sq_powandTauCeti.abs_setIntegral_ball_norm_sub_rpow_mul_div_add_norm_sub_sq_pow_sub_mul_log_le.
References #
- J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987), §3 and §6.
- T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976).
In dimension n ≠ 2, the moment of order 0 ≤ p < n about the centre of the critical bubble
density is integrable for every additive Haar measure.
The moment of order 0 ≤ p < n about the centre of the critical bubble density, in dimension
n ≠ 2, for an additive Haar measure μ. Its dependence on the scale is exactly t ^ p.
Every positive critical-mass moment of order less than the dimension tends to zero as the bubble scale tends to zero from above, for any additive Haar measure.
In dimension n > 2, the moment of order 0 ≤ p < n - 2 about the centre of the
Dirichlet-energy density of a bubble is integrable for every additive Haar measure.
The moment of order 0 ≤ p < n - 2 about the centre of the Dirichlet-energy density of a
bubble, for an additive Haar measure μ. It scales as t ^ p; the excluded endpoint
p = n - 2 is the logarithmic case.
Every positive Dirichlet-energy moment of order less than n - 2 tends to zero as the
bubble scale tends to zero from above, for any additive Haar measure.
Finite moments of the square #
The square moment against an arbitrary real weight rescales by t ^ (p + 2), with the
weight pulled back along z ↦ b + t • z. No integrability or restriction on the moment order or
dimension is required for this identity of totalized integrals.
The moment of order 0 ≤ p < n - 4 of the square of a bubble is integrable for every
additive Haar measure.
The moment of order 0 ≤ p < n - 4 of U² equals
n μ(B(0,1)) 2^(n-3) Β((n+p)/2, (n-p-4)/2) t^(p+2).
The excluded endpoint p = n - 4 is the logarithmic case.
Truncated moments at the logarithmic endpoint #
The moments of order n of U ^ pₙ (in dimension n ≠ 2 with n > 0), of order n - 2 of
|dU|² and of order n - 4 of U² (both in dimension n > 2) diverge logarithmically, and the
moments truncated to a ball are computed instead. Each density is a power of the bubble profile
2 t / (t² + ‖x - b‖²), so these are specializations of
TauCeti.setIntegral_ball_norm_sub_rpow_mul_div_add_norm_sub_sq_pow.
The truncated moment of order n of the critical bubble density, in dimension n ≠ 2, over
the ball B(b, R). It grows like n |B(0, 1)| 2 ^ (n - 1) t ^ n log (1 + (R / t)²), and the
correction is the partial logarithmic series in w = R² / (t² + R²).
The truncated moment of order n of the critical bubble density, in dimension n ≠ 2,
differs from n |B(0, 1)| 2 ^ (n - 1) t ^ n log (1 + (R / t)²) by at most
n |B(0, 1)| 2 ^ (n - 1) t ^ n H_{n - 1}.
The truncated moment of order n - 2 of the Dirichlet-energy density of a bubble, in
dimension n > 2, over the ball B(b, R). It grows like
(n - 2)² n |B(0, 1)| 2 ^ (n - 3) t ^ (n - 2) log (1 + (R / t)²); in dimension 6 this is the
fourth energy moment.
The truncated moment of order n - 2 of the Dirichlet-energy density of a bubble, in
dimension n > 2, differs from (n - 2)² n |B(0, 1)| 2 ^ (n - 3) t ^ (n - 2) log (1 + (R / t)²)
by at most (n - 2)² n |B(0, 1)| 2 ^ (n - 3) t ^ (n - 2) H_{n - 1}.
The truncated moment of order n - 4 of the square of a bubble, in dimension n > 2, over
the ball B(b, R). It grows like n |B(0, 1)| 2 ^ (n - 3) t ^ (n - 2) log (1 + (R / t)²); in
dimension 6 this is the second moment of U².
The truncated moment of order n - 4 of the square of a bubble, in dimension n > 2,
differs from n |B(0, 1)| 2 ^ (n - 3) t ^ (n - 2) log (1 + (R / t)²) by at most
n |B(0, 1)| 2 ^ (n - 3) t ^ (n - 2) H_{n - 3}.
In dimension six, the truncated scalar second moment of a bubble square is an explicit logarithmic term minus a partial logarithmic series.
In dimension six, the scalar second moment differs from its logarithmic leading term
by at most 48 |B(0,1)| t⁴ H₃.