Second Variation Formula and Stability of Exponentially Subelliptic Harmonic Maps

Abstract

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 $\phi : M \rightarrow N$ to the nonlinear PDE system $\tau _b (\phi ) + \phi _*\, \nabla ^H e_b (\phi ) = 0$ [the Euler–Lagrange equations of the variational principle $\delta \, E_b (\phi ) = 0$ where $E_b (\phi ) = \int _\Omega \exp \big [ e_b (\phi ) \big ] \; \Psi$ and $e_b (\phi ) = \frac{1}{2} \, \mathrm{trace}_{G_\theta } \left\{ \Pi _H \phi ^*h \right\}$ and $\Omega \subset M$ is a Carnot–Carathéodory 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)

$$\begin{aligned} H(E_b )_\phi (V, W)= & {} \int _\Omega h^\phi \big ( J^\phi _{b, \, \exp } V, \, W \big ) \; \Psi \\&+\, \int _M \exp \big [ e_b (\phi ) \big ] \, (h^\phi )^*(D^\phi V, \; \Pi _H \phi _*) \, (h^\phi )^*(D^\phi W, \; \Pi _H \phi _*) \; \Psi ,\\ J_{b, \, \exp }^\phi V\equiv & {} \big ( D^\phi \big )^*\big ( \exp \big [ e_b (\phi ) \big ] \; D^\phi V \big ) \\&-\, \exp \big [ e_b (\phi ) \big ] \; \mathrm{trace}_{G_\theta } \left\{ \Pi _H \, \big ( R^h \big )^\phi \big ( V, \; \phi _*\, \cdot \, \big ) \phi _*\cdot \right\} , \end{aligned}$$

[$\Psi = \theta \wedge (d \theta )^n$]. Given a bounded domain $\Omega \subset M$ and an e.s.h. map $\phi \in C^\infty \big ( \overline{\Omega }, \; N \big )$ with values in a Riemannian manifold $N = N^m (k)$ of nonpositive constant sectional curvature $k \le 0$, 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

$$\begin{aligned} \Vert V \Vert _{L^2} \le C \Vert D^\phi V \Vert _{L^2}, \;\; V \in C^\infty _0 \big ( \Omega , \, \phi ^{-1} T N \big ), \end{aligned}$$

and the embedding $\mathring{W}^{1,2}_H (\Omega , \, \phi ^{-1} T N ) \hookrightarrow L^2 (\Omega , \, \phi ^{-1} T N)$ is compact.