Non-rotationally invariant generalised geodesics in the disc
Abstract
We study Brenier's relaxed least-action problem in the unit disc D:={x∈ℝ2: |x|<1} at the critical time T=π, with the endpoint pair iD (the identity on D) and −iD. We prove that the critical energy shell S3={(x,v)∈ℝ2×ℝ2: |x|2+|v|2=1} supports stationary action-minimising generalised incompressible flows that are not invariant under physical rotations. This answers the question posed by Bernot, Figalli, and Santambrogio. Our construction relies on the normalised surface measure σ on S3 and the Hopf quotient Π=(N,M,L):S3→𝕊21/2, 𝕊21/2:={(n,m,ℓ)∈ℝ3: n2+m2+ℓ2=1/4}. Here, N=1/2(x12+v12−x22−v22), M=x1x2+v1v2, L=x1v2−x2v1 . This quotient is a first integral of the harmonic-oscillator flow. Let τ:=Π#σ, let g be a bounded Borel function on 𝕊21/2 whose L∞(τ)-class is odd under (n,m,ℓ)↦(n,m,−ℓ), and let δ∈ℝ. If 1+δg∘Π≥0 σ-a.e., then dμδ,g=π(1+δg∘Π) dσ has Lebesgue spatial marginal and is stationary. The induced path measure is then minimising. We also determine exactly which members of this tilted family are rotationally invariant. If δ≠0, this is equivalent to axisymmetry of the L∞(τ)-class of g, modulo τ-null sets. In particular, taking g(n,m,ℓ)=ℓ m with 0<|δ|<8 gives strictly positive non-rotationally invariant minimisers.
Related
- Geodesics for generalised Monge-Ampère equations
- Finite-rank multiplicative perturbations of rotationally invariant non-Hermitian random matrices
- Spectral properties of deterministic matrices multiplied by rotationally invariant random non-Hermitian ensembles
- Variance lower bounds for geometric functionals of rotationally invariant log-concave random polytopes
- Repeated differentiation of random polynomials with i.i.d. rotationally invariant roots
BibTeX
@misc{gwozdz2026nonrotationally,
title = {{Non-rotationally invariant generalised geodesics in the disc}},
author = {Maja Gwozdz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19008},
eprint = {2609.19008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19008},
}
RIS
TY - UNPB
TI - Non-rotationally invariant generalised geodesics in the disc
AU - Maja Gwozdz
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19008
UR - https://arxiv.org/abs/2609.19008
AB - We study Brenier's relaxed least-action problem in the unit disc D:={x∈ℝ^2: |x|<1} at the critical time T=π, with the endpoint pair i_D (the identity on D) and −i_D. We prove that the critical energy shell S^3={(x,v)∈ℝ^2×ℝ^2: |x|^2+|v|^2=1} supports stationary action-minimising generalised incompressible flows that are not invariant under physical rotations. This answers the question posed by Bernot, Figalli, and Santambrogio. Our construction relies on the normalised surface measure σ on S^3 and the Hopf quotient Π=(N,M,L):S^3→𝕊^2_1/2, 𝕊^2_1/2:={(n,m,ℓ)∈ℝ^3: n^2+m^2+ℓ^2=1/4}. Here, N=1/2(x_1^2+v_1^2−x_2^2−v_2^2), M=x_1x_2+v_1v_2, L=x_1v_2−x_2v_1 . This quotient is a first integral of the harmonic-oscillator flow. Let τ:=Π_#σ, let g be a bounded Borel function on 𝕊^2_1/2 whose L^∞(τ)-class is odd under (n,m,ℓ)↦(n,m,−ℓ), and let δ∈ℝ. If 1+δg∘Π≥0 σ-a.e., then dμ_δ,g=π(1+δg∘Π) dσ has Lebesgue spatial marginal and is stationary. The induced path measure is then minimising. We also determine exactly which members of this tilted family are rotationally invariant. If δ≠0, this is equivalent to axisymmetry of the L^∞(τ)-class of g, modulo τ-null sets. In particular, taking g(n,m,ℓ)=ℓ m with 0<|δ|<8 gives strictly positive non-rotationally invariant minimisers.
ER -