Non-rotationally invariant generalised geodesics in the disc

Abstract

We study Brenier's relaxed least-action problem in the unit disc $D:=\{x\in\mathbb{R}^2:\ |x|<1\}$ 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 \[ S^3=\{(x,v)\in\mathbb{R}^2\times\mathbb{R}^2: |x|^2+|v|^2=1\} \] 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 \[ Π=(N,M,L):S^3\to\mathbb S^2_{1/2}, \qquad \mathbb S^2_{1/2}:=\{(n,m,\ell)\in\mathbb{R}^3: n^2+m^2+\ell^2=1/4\}. \] Here, \[ N=\frac12(x_1^2+v_1^2-x_2^2-v_2^2),\quad M=x_1x_2+v_1v_2,\quad L=x_1v_2-x_2v_1 . \] This quotient is a first integral of the harmonic-oscillator flow. Let $τ:=Π_\#σ$, let $g$ be a bounded Borel function on $\mathbb S^2_{1/2}$ whose $L^\infty(τ)$-class is odd under $(n,m,\ell)\mapsto(n,m,-\ell)$, and let $δ\in\mathbb{R}$. If $1+δg\circΠ\ge0$ $σ$-a.e., then \[ dμ_{δ,g}=π(1+δg\circΠ)\,dσ\] 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 $δ\ne0$, this is equivalent to axisymmetry of the $L^\infty(τ)$-class of $g$, modulo $τ$-null sets. In particular, taking $g(n,m,\ell)=\ell m$ with $0<|δ|<8$ gives strictly positive non-rotationally invariant minimisers.