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–Caratheodory 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–Caratheodory 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 Poincare 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.