Second Variation Formula and Stability of Exponentially Subelliptic Harmonic Maps

We study the stability of exponentially subelliptic harmonic (e.s.h.) maps from a Carnot-Carathéodory complete strictly pseudoconvex pseudohermitian manifold $(M, theta )$ into a Riemannian manifold $(N, h)$. E.s.h. maps are $C^infty$ solutions $phi : M to N$ to the nonlinear PDE system $tau_b (phi ) + phi_ast , nabla^H e_b (phi ) = 0$ [the Euler-Lagrange equations of the variational principle $delta E_b(phi) =0$ where $E_b(phi)=int_Omega exp[ e_b(phi)] Psi$ and $e_b(phi)=(1/2) trace_{G_theta} {Pi_H phi^ast h}$ and $Omega subset M$ is a Carnot-Carath'eodory bounded domain]. We derive the second variation formula about an e.s.h. map, leading to a pseudohermitian analog to the Hessian (of an ordinary exponentially harmonic map between Riemannian manifolds) $H(E_b )_phi (V, W) = int_Omega h^phi ( J^phi_{b, exp} V,W) Psi + int_M exp[e_b(phi )] (h^phi )^ast (D^phi V, Pi_H phi_ast) (h^phi )^ast (D^phi W, Pi_H phi_ast) Psi$ , $J_{b, exp}^phi V equiv (D^phi)^ast ( exp [e_b(phi)] D^phi V) - exp[e_b(phi)] trace_{G_theta}{Pi_H(R^h)^phi (V, phi_ast cdot )phi_ast cdot}$, [$Psi=theta wedge (d theta )^n$]. Given a bounded domain $Omega subset M$ and an e.s.h. map $phi in C^infty (overline{Omega} ,N)$ with values in a Riemannian manifold $N = N^m (k)$ of nonpositive constant sectional curvature $k leq 0$, we solve the generalized Dirichlet eigenvalue problem $J^phi_{b, exp} V=lambda V$ in $Omega$ and $V = 0$ on $partial Omega$ for the degenerate elliptic operator $J^phi_{b, exp}$, provided that $Omega$ supports Poincaré inequality $| V |_{L^2} leq C | D^phi V |_{L^2} , V in C^infty_0 (Omega , phi^{-1} T N)$, and the embedding $IN{W}^{1,2}_H (Omega, phi^{-1} T N) hookrightarrow L^2 (Omega , phi^{-1} T N)$ is compact.