Exponentially subelliptic harmonic maps from the Heisenberg group into a sphere

Abstract

We study critical points $\phi : {{{\mathbb {H}}}}_n \rightarrow S^m$ of the functional

$$\begin{aligned} E_1 (\phi ) = \int _\Omega \exp \left( \frac{1}{2} \, \left\| \nabla ^H \phi \right\| ^2_\theta \right) \; \theta \wedge (d \theta )^n \end{aligned}$$

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

$$\begin{aligned} - \Delta _b \phi ^j + 2 \, e_b (\phi ) \, \phi ^j + G_\theta \big ( \nabla ^H e_b (\phi ) , \, \nabla ^H \phi ^j \big ) = 0, \;\;\; 1 \le j \le m+1, \end{aligned}$$

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 $\phi : {{\mathbb {H}}}_n \rightarrow S^m$ associated to $S^1$ invariant exponentially wave maps $\Phi : C({{\mathbb {H}}}_n ) \rightarrow S^m$ from the total space of the canonical circle bundle $S^1 \rightarrow C({{\mathbb {H}}}_n ) \rightarrow {{\mathbb {H}}}_n$ endowed with the Fefferman metric $F_{\theta }$. We establish Caccioppoli type estimates

$$\begin{aligned} \int _{B_r (x)} \exp \left( \frac{Q}{2} \, \left\| \nabla ^H \phi \right\| ^2_\theta \right) \, \left\| \nabla ^H u \right\| _{\theta }^Q \; \theta \wedge (d \theta )^n \le C r^\beta \end{aligned}$$

($0< \beta < 1$) with $Q = 2 n + 2$ (the homogeneous dimension of ${{\mathbb {H}}}_n$) and show that any weak solution $\phi \in \bigcap _{p \ge Q} W^{1, p}_H (\Omega , S^m )$ of finite p-energy $E_p (\phi ) < \infty$ for some $p \ge 2 Q$ is locally Hölder continuous i.e. $\phi ^j \in S^{0, \alpha }_{\mathrm{loc}} (\Omega )$ for some $0 < \alpha \le 1$ where $S^{0,\alpha } (\Omega )$ are Hölder like spaces, built in terms of the Carnot–Carathéodory metric $\rho _\theta$.