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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 (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.

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Notes

  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.

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Correspondence to Sorin Dragomir.

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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.

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Chiang, YJ., Dragomir, S. & Esposito, F. Second Variation Formula and Stability of Exponentially Subelliptic Harmonic Maps. Complex Anal. Oper. Theory 14, 55 (2020). https://doi.org/10.1007/s11785-020-01012-3

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Keywords

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