Exponentially subelliptic harmonic maps from the Heisenberg group into a sphere

We study critical points $phi: mathbb{H}_n to S^m$ of the functional $$E_1(phi) = int_Omega mathrm{exp} left( frac{1}{2} left | nabla^H phi right |_theta^2 right) theta wedge (d theta )^n$$ for domains $Omega subset subset $mathbb{H}_n$ and a contact structure $theta$ on $mathbb{H}_n$. These are solutions to the second order quasi-linear subelliptic PDE system $$ - Delta_b phi^j + 2 e_b (phi ) phi^l + G_theta( nabla^H e_b (phi ) , nabla^H phi^j ) = 0, 1 leq j leq m+1$$, and arise through Fefferman’s construction (cf. Fefferman in Ann. Math. (2) 103:395–416, 1976; Ann. Math. (2) 104:393–394, 1976) i.e. as base maps $ phi: mathbb{H}_n to S^m$ associated to $S^1$ invariant exponentially wave maps $Phi: C(mathbb{H}_n ) to S^m$ from the total space of the canonical circle bundle $S^1 to C( mathbb{H}_n ) to mathbb{H}_n$ endowed with the Fefferman metric $F_theta$. We establish Caccioppoli type estimates $$ int_{B_r (x)} exp left( frac{Q}{2} left | nabla^H phi right |_theta^2 right) , left | nabla^H right |_theta^Q theta wedge (d theta )^n leq C r^beta$$ ($0 < beta < 1) with $Q = 2n+2$ (the homogeneous dimension of $mathbb{H}_n$) and show that any weak solution $phi in bigcap_{p geq Q} W_H^{1,p} ( Omega, S^m)$ of finite $p$-energy $E_p (phi ) < infty$ for some $p geq 2Q$ is locally Hölder continuous i.e. $phi^j in S_{mathrm{loc}}^{0, alpha} ( Omega )$ for some $0 < alpha leq 1$ where $S^{0, alpha} ( Omega )$ are Hölder like spaces, built in terms of the Carnot–Carathéodory metric $rho_theta$.