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- W3037972142 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) $$begin{aligned} H(E_b )_phi (V, W)= & {} int _Omega h^phi big ( J^phi _{b, , exp } V, , W big ) ; Psi &+, int _M exp big [ e_b (phi ) big ] , (h^phi )^*(D^phi V, ; Pi _H phi _*) , (h^phi )^*(D^phi W, ; Pi _H phi _*) ; Psi J_{b, , exp }^phi Vequiv & {} big ( D^phi big )^*big ( exp big [ e_b (phi ) big ] ; D^phi V big ) &-, exp big [ e_b (phi ) big ] ; mathrm{trace}_{G_theta } left{ Pi _H , big ( R^h big )^phi big ( V, ; phi _*, cdot , big ) phi _*cdot right} , end{aligned}$$[$$Psi = theta wedge (d theta )^n$$]. 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." @default.
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- W3037972142 date "2020-06-24" @default.
- W3037972142 modified "2023-10-08" @default.
- W3037972142 title "Second Variation Formula and Stability of Exponentially Subelliptic Harmonic Maps" @default.
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- W3037972142 doi "https://doi.org/10.1007/s11785-020-01012-3" @default.
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