Skip to main content

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 (Nh). 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.

This is a preview of subscription content, access via your institution.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.


  1. The relevant notions and basic results (of CR and pseudohermitian geometry) are recalled in Sect. 2.1.

  2. Here \(\pi \in {\mathbb R} {\setminus } {\mathbb Q}\) (the irrational number \(\pi \)).

  3. As observed by Valli (Italian mathematician, \(\dag \) 1999, see [21]) at the time [23] was written the e.h. maps theory was quite new and the results in [23] somewhat patchy, yet the adopted expository style made [23] a piece of very enjoyable reading [cf. MR1205818 (94d:58045)].

  4. The result in [20] is about ordinary wave maps, yet the proof of Theorem 4 is a verbatim repetition of the arguments in [20] (hence Theorem 4 is attributed to Duan, cf. op. cit.).

  5. For instance, discreteness of the spectrum of the operator \(J^\phi _b\) associated to a subelliptic harmonic map \(\phi \) is established (cf. [11]) for a class of CR structures arising as orbit spaces \(M^3\) of null Killing vector fields on a space-time (Gödel’s universe in [11]), on a domain \(\Omega \subset M^3\) supporting a form of Poincaré’s inequality and a form of Kondrakov compactness involving \(L^2 (\Omega , \, \phi ^{-1} T N)\). The approach in [11] carries over verbatim to arbitrary subelliptic harmonic maps.

  6. That is the assumptions in Theorem 8, including the curvature requirements on the Riemannian manifold (Nh), together with the Kondrakov condition.


  1. Ara, M.: Geometry of \(F\)-harmonic maps. Kodai Math. J. 22(2), 243–263 (1999)

    MathSciNet  Article  Google Scholar 

  2. Ara, M.: Stability of \(F\)-harmonic maps into pinched manifolds. Hiroshima Math. J. 31(1), 171–181 (2001)

    MathSciNet  Article  Google Scholar 

  3. Ara, M.: Instability and nonexistence theorems for F-harmonic maps. Ill. J. Math. 45(2), 657–679 (2001)

    MathSciNet  Article  Google Scholar 

  4. Aribi, A., Dragomir, S., El Soufi, A.: On the continuity of the eigenvalues of a sublaplacian. Can. Math. Bull. 57(1), 12–24 (2014)

    MathSciNet  Article  Google Scholar 

  5. Aribi, A., Dragomir, S., El Soufi, A.: Eigenvalues of the sublaplacian and deformations of contact structures on a compact CR manifold. Differ. Geom. Appl. 39, 113–128 (2015)

    Article  Google Scholar 

  6. Aronsson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6(28), 551–561 (1966)

    MathSciNet  MATH  Google Scholar 

  7. Barletta, E.: Subelliptic \(F\)-harmonic maps. Riv. Mat. Parma 2(7), 33–50 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Barletta, E., Dragomir, S., Urakawa, H.: Pseudoharmonic maps from nondegenerate CR manifolds to Riemannian manifolds. Indiana Univ. Math. J. 50, 719–746 (2001)

    MathSciNet  Article  Google Scholar 

  9. Barletta, E., Dragomir, S., Jacobowitz, H., Soret, M.: \(b\)-completion of pseudo-Hermitian manifolds. Class. Quantum Gravity 29, 095007 (27 pp) (2012)

    MathSciNet  Article  Google Scholar 

  10. Barletta, E., Dragomir, S., Jacobowitz, H.: Gravitational field equations on Fefferman space-times. Complex Anal. Oper. Theory 11, 1685–1713 (2017)

    MathSciNet  Article  Google Scholar 

  11. Barletta, E., Dragomir, S., Magliaro, M.: Wave maps from Gödel’s universe. Class. Quantum Gravity 39(19), 195001 (52 pp) (2014)

    MATH  Google Scholar 

  12. Bony, M.: Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés. Ann. Inst. Fourier Grenoble 1(19), 277–304 (1969)

    Article  Google Scholar 

  13. Boutet de Monvel, L.: Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974-1975; Équations aux derivées partielles linéaires et non linéaires, pp. Exp. No. 9, 14 pp. Centre Math., École Polytech., Paris (1975)

  14. Chiang, Y.-J.: Exponentially harmonic maps and their properties. Math. Nachr. 288(17–18), 1970–1980 (2015)

    MathSciNet  Article  Google Scholar 

  15. Chiang, Y.-J., Yang, Y.: Exponential wave maps. J. Geom. Phys. 57(12), 2521–2532 (2007)

    MathSciNet  Article  Google Scholar 

  16. Chiang, Y.-J., Dragomir, S., Esposito, F.: Exponentially subelliptic harmonic maps from the Heisenberg group into a sphere. Calc. Var. Partial Differ. Equ. 58, 125 (2019)

    MathSciNet  Article  Google Scholar 

  17. Danielli, D., Garofalo, N., Nhieu, D-M.: Non doubling Ahlfors measures, perimeter measures, the characterization of the trace spaces of Sobolev functions in Carnot–Carathéodory spaces. Mem. Am. Math. Soc. 182(857) (2006)

  18. Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, vol. 246. Birkhäuser, Boston (2006)

    MATH  Google Scholar 

  19. Dragomir, S., Perrone, D.: Levi harmonic maps of contact Riemannian manifolds. J. Geom. Anal. 24(3), 1233–1275 (2014)

    MathSciNet  Article  Google Scholar 

  20. Duan, Y.: Harmonic maps and their application to general relativity. SLAC-PUB-3265, December 1983 (T), Stanford Linear Accelerator Center, Stanford, CA (unpublished)

  21. Eells, J.: On the mathematical contribution of Giorgio Valli. Rendiconti di Matematica, Ser. VII, vol. 22, Roma, pp. 147–158 (2002)

  22. Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)

    MathSciNet  Article  Google Scholar 

  23. Eells, J., Lemaire, L.: Some properties of exponentially harmonic maps. In: Partial Differential Equations, Banach Center Publications, vol. 27, Institute of Mathematics, Polish Academy of Sciences, Warszawa, pp. 129–136 (1992)

  24. Graham, C.R.: On Sparling’s characterization of Fefferman metrics. Am. J. Math. 109, 853–874 (1987)

    MathSciNet  Article  Google Scholar 

  25. Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    MathSciNet  Article  Google Scholar 

  26. Lee, J.M.: The Fefferman metric and pseudohermitian invariants. Trans. A.M.S. 296(1), 411–429 (1986)

    MATH  Google Scholar 

  27. Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. R. Soc. Lond. A 264, 413–496 (1969)

    MathSciNet  Article  Google Scholar 

  28. Menikoff, A., Sjöstrand, J.: On the eigenvalues of a class of hypoelliptic operators. Math. Ann. 235, 55–58 (1978)

    MathSciNet  Article  Google Scholar 

  29. Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91, 259, ISBN: 978-0-8218-4165-5 (2002)

  30. Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. Math. 113, 1–24 (1981)

    MathSciNet  Article  Google Scholar 

  31. Smith, R.T.: The second variation formula for harmonic mappings. Proc. Am. Math. Soc. 47(1), 229–236 (1975)

    MathSciNet  Article  Google Scholar 

  32. Strichartz, R.S.: Sub-Riemannian geometry. J. Differ. Geom. 24, 221–263 (1986)

    MathSciNet  Article  Google Scholar 

  33. Jost, J., Xu, C.-J.: Subelliptic harmonic maps. Trans. Am. Math. Soc. 350(11), 4633–4649 (1998)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Sorin Dragomir.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.

Communicated by Fabrizio Colombo.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chiang, YJ., Dragomir, S. & Esposito, F. Second Variation Formula and Stability of Exponentially Subelliptic Harmonic Maps. Complex Anal. Oper. Theory 14, 55 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI:


  • CR manifold
  • Tanaka–Webster connection
  • Fefferman’s metric
  • Exponentially subelliptic harmonic map
  • Stability