Abstract
We determine all proper holomorphic maps of balls \({\mathbb {B}}_2 \rightarrow {\mathbb {B}}_3\) admitting a \(C^3\) extension up to the boundary of \({\mathbb {B}}_2\) and whose boundary values \(S^3 \rightarrow S^5\) are subelliptic harmonic maps (in the sense of Jost and Xu in Trans Am Math Soc 350(11):4633–4649, 1998). A new numerical CR invariant, the CR degree of a CR map of spheres \(S^{2n+1} \rightarrow S^{2N+1}\), is introduced and used to distinguish among the spherical equivalence classes in Faran’s list \(P^*(2,3)\) (cf. Faran in Invent Math 68:441–475, 1982). As an application, the boundary values \(\phi \) of Alexander’s map \(\Phi \in P(2,3)\) (cf. Alexander in Indiana Univ Math J 26:137–146, 1977) is shown to be homotopically nontrivial, as a map of \(\{ (z,w) \in S^3 : w + \overline{w} > 0 \}\) into \(S^5 {\setminus } S^3\).
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1 A glimpse to the main results
By a classical result of Webster [34], if \(\Phi : {\mathbb {B}}_n \rightarrow {\mathbb {B}}_{n+1}\) is a proper holomorphic map which is \(C^3\) up to the boundary of \({\mathbb {B}}_n\) and \(n \ge 3\), then \(\Phi \) is linear fractional. When \(n = 2\) the previous statement is false, as shown for instance by the example \(\Phi (z,w) = ( z^2 , \, \sqrt{2} \, z w , \, w^2 )\) (due to Alexander [1]). Instead, proper holomorphic maps \(\Phi : {\mathbb {B}}_2 \rightarrow {\mathbb {B}}_3\) which are \(C^3\) up to the boundary were classified by Faran [18], up to spherical equivalence. Two maps \(\Phi , \, \varPsi : {\mathbb {B}}_2 \rightarrow {\mathbb {B}}_3\) are spherically equivalent if \(\varPsi = \zeta \circ \Phi \circ \xi ^{-1}\) for some \(\xi \in \mathrm{Hol}({\mathbb {B}}_2 )\) and \(\zeta \in \mathrm{Hol} ({\mathbb {B}}_3 )\). Let \(O(2,3)\) be the set of all proper holomorphic maps from \({\mathbb {B}}_2\) into \({\mathbb {B}}_3\). Let \(P(2, 3)\) consist of all \(\Phi \in O(2,3)\) such that \(\Phi \) extends holomorphically past the boundary of \({\mathbb {B}}_2\) and let \(P^*(2,3)\) be the corresponding quotient space, modulo spherical equivalence. Faran’s result is that \(P^*(2,3)\) consists precisely of four classes \(\{ {\mathbb {F}}, \; {\mathbb {A}}_0 , \; {\mathbb {A}}_1 , \; {\mathbb {I}} \}\) and the following maps are representatives
for any \((z,w) \in {\mathbb {B}}_2\). If \({\mathcal {C}} \in P^*(2,3)\) then let \({\mathcal {C}}_b\) denote its boundary values i.e., \({\mathcal {C}}\) is the set of all maps \(\phi : S^3 \rightarrow S^5\) such that \(j_2 \circ \phi = \Phi \circ j_1\) as \(\Phi \) ranges over \(\mathcal {C}\). Here \(j_n : S^{2n+1} \rightarrow {\mathbb {C}}^{n+1}\) is the inclusion. Our purpose through this paper is to distinguish among the four classes \({\mathcal {C}} \in P^*(2,3)\), from the point of view of the geometry of the second fundamental form of the maps in \({\mathcal {C}}_b\). To this end, we look for the subelliptic harmonic (in the sense of Jost and Xu [23]) representatives of each \({\mathcal {C}}_b\). Our main finding (an abridged version of Theorem 1 in Sect. 3) is
Corollary 1
The maps \(U \circ \Phi _{{\mathbb {A}}_0} \circ u^{-1}\) and \(U \circ \Phi _{\mathbb {I}} \circ u^{-1}\) as \(u \in \mathrm{U}(2)\) and \(U \in \mathrm{U}(3)\) are all the proper holomorphic maps \({\mathbb {B}}_2 \rightarrow {\mathbb {B}}_3\) admitting a \(C^3\) extension to the boundary of \({\mathbb {B}}_2\) whose boundary values \(S^3 \rightarrow S^5\) are subelliptic harmonic as maps of the pseudohermitian manifold \((S^3 , \, \eta _0 )\) into the Riemannian manifold \((S^5 , \, g_{E_0} )\). Every subelliptic harmonic map in the above list is unstable.
Here \(\eta _0 = j_1^*\left\{ \frac{i}{2} \left( \overline{\partial } - \partial \right) \left( |z|^2 + |w|^2 \right) \right\} \) and \(E_0 = j_2^*\left\{ \frac{i}{2} \left( \overline{\partial } - \partial \right) |Z|^2 \right\} \). Also \(g_{E_0}\) is the Webster metric associated to the contact form \(E_0\) (the standard Sasakian metric on \(S^5\)). As another manner to understand the nature of maps in \(P (2,3)\), inspired by the theory of topological degree, we introduce a new numerical CR invariant of \(S^{2n+1}\) associated to every CR map \(\phi : S^{2n+1} \rightarrow S^{2N+1}\) (with \(N = n + k\), \(k \ge 0\)), i.e.,
referred to as the CR degree of \(\phi \). Here \(\eta \) is an arbitrary positively oriented contact form on \(S^{2n+1}\). Also \(\omega _n\) is the volume of \(S^{2n+1}\) with respect to the volume form \(\varPsi _{\eta _0} = \eta _0 \wedge (\mathrm{d}\eta _0 )^n\) and \(\lambda (\phi ; \, \eta , \, E_0 )\) is the dilation of \(\phi \) as a map of contact manifolds \((S^3 , \eta )\) and \((S^5 , E_0 )\). If \(f \in L^p (S^{2n+1} )\) and \(1 \le p < \infty \) then let \(\Vert f \Vert _p = ( \int _{S^{2n+1}} |f|^p \, \varPsi _{\eta })^{1/p}\) be the \(L^p\) norm of \(f\). Then \(\Vert \lambda (\phi ; \, \eta , \, E_0 ) \Vert _p\) is a CR invariant of \(S^{2n+1}\) if and only if \(p = n+1\). The CR degree of a CR map \(\phi : S^{2n+1} \rightarrow S^{2N+1}\) coincides with its topological degree in the equidimensional (i.e., \(k = 0\)) case. Given a polynomial holomorphic map \(\Phi = (\Phi _1 , \ldots , \Phi _N ) : {\mathbb {B}}_n \rightarrow {\mathbb {B}}_N\) we denote by \(p(\Phi )\) the maximum of the degrees of the polynomials \(\Phi _j\) (\(1 \le j \le N\)). We show that (cf. Theorem 2 in Sect. 5)
Corollary 2
For every \({\mathcal {C}} \in P^*(2,3)\) there is a constant \(0 < \varLambda _{\mathcal {C}} \le 1\) such that
with \(\varLambda _{\mathcal {C}} = 1\) when \(\phi _{\mathcal {C}}\) is subelliptic harmonic and \(\varLambda _{\mathcal {C}} < 1\) otherwise. CR degree is not a homotopy invariant. If \({\mathcal {C}} \in P^*(2,3)\) and \(\Phi \in {\mathcal {C}}\) then \(\mathrm{deg}(\phi ) \approx \mathrm{deg} (\phi _{\mathcal {C}})\). Precisely if \(\Phi = \zeta \circ \Phi _{\mathcal {C}} \circ \xi ^{-1}\) with \(\zeta = U \circ \varphi _A\), \(\xi = u \circ \varphi _a\), \(U \in \mathrm{U}(3)\), \(u \in \mathrm{U}(2)\), \(A \in {\mathbb {B}}_3\), \(a \in {\mathbb {B}}_2\), then
where \(C_{a,A} = (1 + |a|)^4 (1 - |a|)^{-4} (1 + |A|)^2 (1 - |A|)^{-2}\).
By a result of Solomon [27], any harmonic map \(\phi : M \rightarrow S^m\) of a compact Riemannian manifold \(M\) into a sphere \(S^m\), which omits a codimension two totally geodesic submanifold \(\Sigma \subset S^m\), is homotopically nontrivial. The boundary values \(\phi _{{\mathbb {A}}_0} : M = \{ (z,w) \in S^3 : \mathrm{Re} (w) > 0 \} \rightarrow S^5\) of Alexander’s map \(\Phi _{{\mathbb {A}}_0}\) is a subelliptic harmonic map omitting \(S^3\). We show (cf. Theorem 3 in Sect. 6) that
Corollary 3
\(\phi _{{\mathbb {A}}_0} : M \rightarrow S^5 {\setminus } S^3\) links \(S^3\).
2 Holomorphic and harmonic maps
By a classical result of Lichnerowicz (cf. [24]), every holomorphic map of closed (i.e., compact, without boundary) Kähler manifolds is harmonic and an absolute minimum to the Dirichlet energy functional within its homotopy class. That holomorphic maps of complex manifolds are harmonic with respect to any choice of Kählerian metrics, but an elementary result in differential geometry. The stability part in A. Lichnerowicz’s theorem (cf. op. cit.) is more subtle and relies on a deformation argument requiring compactness. Arguments in the proof appear as confined to Kählerian geometry and rigidly tied to even dimensionality. For instance, if \(\Phi : M \rightarrow N\) is a holomorphic map of compact generalized Hopf manifolds (i.e., locally conformal Kähler manifolds with a parallel Lee form, cf. e.g., [11], p. 33) then \(\Phi \) remarkably is (by a result of Tsukada [29, 30]) a foliated map (with respect to the canonical foliations \({\mathcal F}_M\) and \({\mathcal F}_N\) by Riemann surfaces, as discovered by Vaisman [32]) yet only transversally harmonic, or \(({\mathcal F}_M , {\mathcal F}_N )\)-harmonic (notably \({\mathcal F}_M\) and \({\mathcal F}_N\) are transversally Kählerian) and weakly stable (cf. Barletta et al. [3]; Tommasoli et al. [14]). Also, in the odd dimensional case, every CR map \(\phi : M \rightarrow N\) of compact Sasakian manifolds, preserving the Reeb flows \({\mathcal F}\) and \({\mathcal G}\) of \(M\) and \(N\), respectively, is a weakly stable \(({\mathcal F}, {\mathcal G})\)-harmonic map (cf. Theorem 8 in [15]).
Let \({\mathbb {B}}_n = \{ z \in {\mathbb {C}}^n : |z| < 1 \}\) be the standard unit ball. The purpose of the present paper is to study holomorphic maps of balls \(\Phi : {\mathbb {B}}_n \rightarrow {\mathbb {B}}_N\), from the point of view of harmonic map theory. As emphasized above, any such map is harmonic, regardless of the Kählerian metrics fixed on \({\mathbb {B}}_n\) and \({\mathbb {B}}_N\), yet Lichnerowicz’s theorem doesn’t apply for lack of compactness. Neverthless if \(\Phi \) extends smoothly to the boundary and \(\Phi \left( S^{2n-1} \right) \subset S^{2N-1}\) then its boundary values \(\phi : S^{2n-1} \rightarrow S^{2N-1}\) is a CR map. It is a natural question whether \(\phi \) is a subelliptic harmonic map (in the sense of Jost and Xu [23]; Barletta et al. [4], cf. definitions in Sect. 3) with respect to the canonical pseudohermitian structure on \(S^{2n-1}\) and the standard Sasakian metric on \(S^{2N-1}\). If this is the case, \(\phi \) is a map of compact manifolds and a stability theory (following the lines of stability theory for ordinary harmonic maps, cf. e.g., Urakawa [31]) may be built (cf. our Sect. 4 below).
The previous question may be labeled as “natural” due to the following instance. Let \(\varOmega \subset {\mathbb {C}}^n\) (\(n \ge 2\)) be a smoothly bounded strictly pseudoconvex domain and \(S\) a Riemannian manifold. Let \(\Phi : \varOmega \rightarrow S\) be a Bergman-harmonic map, i.e., \(\Phi \) is \(C^\infty \) and harmonic as a map among the Riemannian manifolds \((\varOmega , g_\varOmega )\) and \(S\), where \(g_\varOmega \) is the Bergman metric on \(\varOmega \). Assume that \(\Phi \) extends smoothly to the boundary and \(\lim _{\varOmega \ni z \rightarrow z_0} \Phi (z) \in S\) for every \(z_0 \in \partial \varOmega \), and let \(\phi : \partial \varOmega \rightarrow S\) be its boundary values. If the pullback \(\Phi ^{-1} \nabla ^S\) of the Levi-Civita connection \(\nabla ^S\) of \(S\) stays bounded at the boundary (precisely, for every local coordinate system \((V, y^\alpha )\) on \(S\) such that \(U = \Phi ^{-1} (V)\) is a one sided neighborhood of (a portion of) \(\partial \varOmega \) one has \(\lim _{U \ni z \rightarrow z_0} \varGamma ^\alpha _{\beta \gamma } (\Phi (z)) = \varGamma ^\alpha _{\beta \gamma } (\Phi (z_0 ))\) for every \(z_0 \in \partial U \cap \partial \varOmega \), where \(\varGamma ^\alpha _{\beta \gamma }\) are the Christoffel symbols of \(\nabla ^S\) relative to \((V, y^\alpha )\)) and \(\phi \) has vanishing normal derivatives then (by a result in [5]) \(\phi \) is a subelliptic harmonic map.
It should be mentioned, however, that the ball \(S = {\mathbb {B}}_N\), thought of as carrying the Bergman metric \(g_{{\mathbb {B}}_N}\), doesn’t satisfy the assumption above (\(\Phi ^{-1} \nabla ^{{\mathbb {B}}_N}\) is infinite at the boundary) and a rather different boundary behavior may be observed: one has \(\Vert \Phi (z) \Vert \rightarrow 1\) as \(\varOmega \ni z \rightarrow \partial \varOmega \) and the boundary values of the (Bergman-harmonic) map \(\Phi : \varOmega \rightarrow {\mathbb {B}}_N\) may be shown (cf. [5]) to satisfy a first-order PDE system similar to the tangential Cauchy–Riemann equations (and satisfied by any CR map \(\phi : \partial \varOmega \rightarrow S^{2N-1}\)).
A result in this paper is that a CR map \(\phi : M \rightarrow S\) of strictly pseudoconvex CR manifolds, endowed with positively oriented contact forms \(\theta \) and \(\Theta \), respectively, is subelliptic harmonic as a map of the pseudohermitian manifold \((M, \theta )\) into the Riemannian manifold \((S, g_\Theta )\) (where \(g_\Theta \) is the Webster metric of \((S, \Theta )\)) if and only if the dilation \(\lambda (\phi ) = \lambda (\phi , \theta , \Theta )\) is a constant and \(\lambda (\phi ) T_\Theta ^\phi - \phi _*T_\theta = 0\). Here \(T_\theta \) and \(T_\Theta \) are the Reeb vector fields of \((M, \theta )\) and \((S, \Theta )\), respectively. Loosely speaking, subelliptic harmonicity of CR maps \(\phi : M \rightarrow S\) may be characterized in terms of the position occupied by \(\phi (M)\) with respect to the ambient “preferential direction” \(T_\Theta \). If for instance \(\phi \) is a CR immersion then a necessary condition for subelliptic harmonicity is that the Reeb field \(T_\Theta \) be tangent to the submanifold \(\phi (M)\). This may be used as a tool for searching for subelliptic harmonic representatives in the boundary values \({\mathcal {C}}_b\) of each spherical equivalence class \({\mathcal {C}} \in P^*(2, 3)\) in Faran’s list.
A stability theory confined, for our needs in this paper, to subelliptic harmonic maps of the sphere \(S^3\), endowed with the canonical positively oriented contact form \(\eta _0\), into a Riemannian manifold, is built in Sect. 4 and its outcomes are used to prove the last statement in Corollary 1. As in stability theory for ordinary harmonic maps, the corner stone is the second variation formula
(cf. Sect. 4 below). Consequently, we may introduce pseudohermitian analogs \(\mathrm{null}_b (\phi )\) and \(\mathrm{ind}_b (\phi )\) to nullity and index of the Hessian of the energy functional at a subelliptic harmonic map \(\phi : S^3 \rightarrow S\). The proof of the instability statement in Corollary 1 is then to compute \(\mathrm{Hess}_b (E)_\phi (V_j , V_j )\) where \(V_j = \mathrm{tan} (Y_j )^\phi \in C^\infty (\phi ^{-1} T(S^5 ))\) for some globally defined parallel orthonormal frame \(\{ Y_j \}\) on \({\mathbb {C}}^3\) and exploit the fact that the standard Riemannian metric \(g_{E_0}\) on \(S^5\) has positive sectional curvature.
A basic problem in harmonic map theory is, given a homotopy class \({\mathcal {C}} \in \pi _0 (M,S)\) of maps among Riemannian manifolds \(M\) and \(S\), to determine a harmonic representative of \({\mathcal {C}}\). A solution is known when \(S\) has nonpositive sectional curvature (cf. [16]). Also (by a result in [17]) there is a Riemannian metric \(\hat{g}\) on \(M\), conformal to the original metric, and there is \(\phi \in {\mathcal {C}}\) which is harmonic as a map of \((M, \, \hat{g})\) into \(S\). The natural pseudohermitian analog to harmonic maps has been devised in [4] as a global notion (originally termed pseudoharmonic maps) one of whose local manifestations was discovered independently in [23], i.e., the subelliptic harmonic maps. The result in [16] was recently recovered by Zhou [35] (deforming a map \(\phi \in {\mathcal {C}}\) into a subelliptic harmonic map through the heat flow). No subelliptic analog to the result in [17] is known so far. As announced, we take up the analogous problem of determining a subelliptic harmonic representative in the boundary values \({\mathcal {C}}_b\) of each of spherical equivalence class \({\mathcal {C}} \in P^*(2, 3)\). The basics of subelliptic harmonic maps theory are recalled in Sect. 3 where we also prove part of Theorem 1. The proof of Theorem 1 is completed in Sect. 4 (showing that \(\phi _{{\mathbb {A}}_0}\) and \(\phi _{\mathbb {I}}\) are unstable).
3 CR and subelliptic harmonic maps
Let \((M, T_{1,0}(M))\) be an orientable CR manifold, of CR dimension \(n\), where \(T_{1,0}(M) \subset T(M) \otimes {\mathbb {C}}\) denotes its CR structure. The Levi, or maximally complex, distribution is \(H(M) = \mathrm{Re} \left\{ T_{1,0}(M) \oplus T_{0,1}(M) \right\} \) where \(T_{0,1}(M) = \overline{T_{1,0}(M)}\). It carries the complex structure \(J : H(M) \rightarrow H(M)\) given by \(J(Z + \overline{ Z}) = i (Z - \overline{Z})\) for every \(Z \in T_{1,0}(M)\). Let \(H(M)^\bot \subset T^*(M)\) be the conormal bundle associated to \(H(M)\), i.e.,
A pseudohermitian structure is a globally defined nowhere zero section \(\theta \in C^\infty (H(M)^\bot )\). The Levi form is \(G_\theta (X, Y) = (\mathrm{d}\theta ) (X , J Y)\) for any \(X,Y \in H(M)\). If \(\hat{\theta }\) is another pseudohermitian structure on \(M\) then \(\hat{\theta } = \lambda \theta \) for some \(C^\infty \) function \(\lambda : M \rightarrow {\mathbb {R}} {\setminus } \{ 0 \}\) and consequently the Levi form changes conformally \(G_{\hat{\theta }} = \lambda G_\theta \). A pair \((M, \theta )\) is a pseudohermitian manifold. A CR manifold is nondegenerate (respectively, strictly pseudoconvex) if the Levi form \(G_\theta \) is nondegenerate (respectively, positive definite) for some \(\theta \). Let \(\mathcal P\) be the set of all pseudohermitian structures on \(M\). If \(M\) is nondegenerate then each \(\theta \in {\mathcal P}\) is a contact form, i.e., \(\varPsi _\theta = \theta \wedge (\mathrm{d}\theta )^n\) is a volume form on \(M\). Also there is a unique globally defined nowhere zero tangent vector field \(T \in \mathfrak {X}(M)\) (the Reeb vector of \((M, \theta )\)) such that \(\theta (T) = 1\) and \(T \, \rfloor \, \mathrm{d}\theta = 0\). In particular \(T\) is transverse to the Levi distribution. It is customary to extend \(J\) to a \((1,1)\)-tensor field on \(M\) by requesting that \(J T = 0\) (and then \(J\) becomes an \(f\)-structure on \(M\), i.e., \(J^2 = - I + \theta \otimes T\)). If \(M\) is strictly pseudoconvex, then \(\mathcal P\) splits into two equivalence class \({\mathcal P}_\pm \) as the corresponding Levi form is positive or negative definite. Contact forms \(\theta \in {\mathcal P}_+\) are referred to as positively oriented. Notations and conventions on CR and pseudohermitian geometry are those in [14]. A \(C^\infty \) map of CR manifolds \(\phi : M \rightarrow S\) is a CR map if \((d_x \phi ) T_{1,0}(M)_x \subset T_{\phi (x)} (S)\) for every \(x \in M\). Equivalently \(\phi \) is a CR map if \((d_x \phi ) H(M)_x \subset H(S)_{\phi (x)}\) and \(J_{S, \phi (x)} \circ (d_x \phi ) = (d_x \phi ) \circ J_x\) on \(H(M)_x\) for every \(x \in M\). Here \(J_S\) denotes the complex structure on \(H(S)\). A CR isomorphism is a \(C^\infty \) diffeomorphism and a CR map. \(\mathrm{Aut}_\mathrm{CR}(M)\) denotes the group of all CR isomorphisms of \(M\) into itself. CR maps are natural CR analogs to holomorphic maps in complex analysis. Elementary evidence is that for every holomorphic map \(\Phi : U \rightarrow V\) among open sets \(U \subset {\mathbb {C}}^{n+1}\) and \(V \subset {\mathbb {C}}^{N+1}\), preserving given real hypersurfaces \(M \subset U\) and \(S \subset V\), i.e., \(\Phi (M) \subset S\), its trace \(\phi : M \rightarrow S\) is a CR map. Here \(M\) and \(S\) are endowed with the CR structure induced by the complex structure of the ambient space, e.g.,
Deeper evidence is that, by a classical result of Fefferman [19], any biholomorphism \(\Phi \in \mathrm{Hol}(\varOmega )\) of a smoothly bounded strictly pseudoconvex domain \(\varOmega \subset {\mathbb {C}}^{n+1}\) extends smoothly to a CR isomorphism \(\phi \in \mathrm{Aut}_\mathrm{CR}(M)\) of the boundary \(M = \partial \varOmega \).
Let \(M\) be a strictly pseudoconvex CR manifold and \(\theta \in {\mathcal P}_+\) a positively oriented contact form on \(M\). The Webster metric is the Riemannian metric \(g_\theta \) on \(M\) given by
for any \(X,Y \in H(M)\). Let \(S\) be a Riemannian manifold, with the Riemannian metric \(h\). A \(C^\infty \) map \(\phi : M \rightarrow S\) is subelliptic harmonic if it is a critical point of the energy functional
for every relatively compact domain \(\varOmega \subset M\). If \(B\) is a bilinear form on \(T(M)\) then \(\varPi _H \, B\) denotes its restriction to \(H(M) \otimes H(M)\). A map \(\phi \in C^\infty (M, S)\) is a critical point of \(E_\varOmega \) if
for every smooth 1-parameter variation \(\{ \phi _t \}_{|t| < \epsilon } \subset C^\infty (M, S)\) of \(\phi \) (i.e., \(\phi _0 = \phi \)) such that \(\mathrm{Supp}(V) \subset \varOmega \). Here \(V \in C^\infty (\phi ^{-1} T(S))\) is the infinitesimal variation induced by \(\{ \phi _t \}_{|t| < \epsilon }\) i.e.,
Assume that \(M\) is compact and set \(E = E_M\) for simplicity. Energy \(E\) has been discovered (cf. [4]) by integrating along the fibers in the Dirichlet integral
where \(C(M)\) is the total space of the canonical circle bundle \(S^1 \rightarrow C(M) \mathop {\longrightarrow }\limits ^{p} M\) (cf. [14], p. 119) and \(F_\theta \) is the Fefferman metric of \((M, \theta )\) (a Lorentzian metric on \(C(M)\), cf. Definition 2.15 in [14], p. 128). Indeed, as it turns out, \({\mathbb {E}} (\phi \circ p) = 2 \pi \, E(\phi )\). Critical points of \({\mathbb {E}} : C^\infty (C(M), S) \rightarrow {\mathbb {R}}\) are ordinary harmonic maps of \((C(M), F_\theta )\) into \((S, h)\). Source manifold is, as emphasized above, but Lorentzian yet the theory of harmonic maps within the semi-Riemannian category is already consolidated (cf. work by Fuglede [21]). The first variation formula for \(E\) is (cf. [4])
where \(\tau _b (\phi ) \in C^\infty (\phi ^{-1} T(S))\) is
Also \(h^\phi = \phi ^{-1} h\) is the pullback of \(h\) (a Riemannian bundle metric in the pullback bundle \(\phi ^{-1} T(S) \rightarrow M\)). The \((1,2)\)-tensor field \(\beta _b (\phi )\) is the (pseudohermitian analog to the) second fundamental form of \(\phi \) (discovered by Petit [25]). As in the ordinary second fundamental form (cf. [2], p. 69) \(\left( \nabla ^S \right) ^\phi = \phi ^{-1} \nabla ^S\) is the pullback of the Levi-Civita connection \(\nabla ^S\) of \((S, h)\) (a connection in \(\phi ^{-1} T(S) \rightarrow M\) parallelizing the bundle metric \(h^\phi \)) yet the Levi-Civita connection of the source manifold is replaced by the Tanaka-Webster connection \(\nabla ^\theta \) of \((M, \theta )\). As well known, the existence of \(\nabla ^\theta \) requires (cf. [28, 33]) only nondegeneracy of the given CR manifold \(M\) and may be characterized as the unique linear connection on \(M\), associated to a fixed contact form \(\theta \in {\mathcal P}\), satisfying the following axioms i) \(H(M)\) is parallel with respect to \(\nabla ^\theta \), ii) \(\nabla ^\theta J = 0\) and \(\nabla ^\theta g_\theta = 0\), iii) \(T_{\nabla ^\theta } (Z, W) = 0\) and \(T_{\nabla ^\theta } (Z, \overline{W}) = 2 i G_\theta (Z, \overline{W}) T\) for any \(Z,W \in T_{1,0}(M)\) and iv) \(\tau _\theta \circ J + J \circ \tau _\theta = 0\). Here \(T_{\nabla ^\theta }\) is the torsion tensor field of \(\nabla ^\theta \) and \(\tau _\theta (X) = T_{\nabla ^\theta } (T, X)\) for every \(X \in \mathfrak {X}(M)\) (the pseudohermitian torsion of \(\nabla ^\theta \)).
Let \(\overline{\partial }_b\) be the tangential Cauchy–Riemann operator, i.e., \((\overline{\partial }_b f) \overline{Z} = \overline{Z}(f)\) for every \(C^1\) function \(f : M \rightarrow {\mathbb {C}}\) and any \(Z \in T_{1,0}(M)\). Then \(\overline{\partial }_b f = 0\) are the tangential Cauchy-Riemann equations and a \(C^1\) solution \(f\) is a CR function. Any real-valued CR function \(u\) is a constant. Indeed, let \(\{ T_\alpha : 1 \le \alpha \le n \}\) be a local frame of \(T_{1,0}(M)\), defined on the open set \(U \subset M\), and let us set \(\nabla ^\theta _{T_A} T_B = \varGamma _{AB}^C T_C\) for some \(C^\infty \) funcitons \(\varGamma _{AB}^C : U \rightarrow {\mathbb {C}}\), where \(A,B,C, \ldots \in \{ 0, \, 1, \ldots , n, \, \overline{1}, \ldots , \overline{n} \}\) and \(T_{\overline{\alpha }} = \overline{T}_\alpha \) and \(T_0 = T\). As \(\overline{\partial }_b u = 0\) one has \(T_{\overline{\nu }} (u) = 0\) hence (by complex conjugation) \(T_\nu (u) = 0\). By axioms (ii)–(iii) above
where \(g_{\mu \overline{\nu }} = G_\theta (T_\mu , T_{\overline{\nu }})\), so that \(T(u) = 0\). It follows that \(u\) is locally constant and then constant (\(M\) is tacitly connected).
So \(\tau _b (\phi )\) is a pseudohermitian analog to the tension field in harmonic map theory (cf. e.g., Definition 3.2.4 in [2], p. 71) and \(\phi \in C^\infty (M, S)\) is subelliptic harmonic if and only if \(\tau _b (\phi ) = 0\). Note that \(\tau _b (\phi )\) is the trace, with respect to the Levi form \(G_\theta \), of the restriction of \(\beta _b (\phi )\) to \(H(M) \otimes H(M)\) (rather than the trace of the full \(\beta _b (\phi )\)). That this is a quite general procedure, leading to degenerate elliptic systems of PDEs (whose ellipticity degenerates precisely in the missed direction \(T\)) was emphasized by Perone et al. [12], and ramifications of this phenomenon (for maps from contact Riemannian manifolds, with a generally nonintegrable almost CR structure, cf. [6]) were recently investigated by Petit et al. [13]. Indeed, \(\tau _b (\phi ) = 0\) may be locally written as the quasi-linear subelliptic PDEs system
where \(\phi ^\alpha = y^\alpha \circ \phi \), \(\Delta _b\) is the second-order differential operator (the sublaplacian of \((M, \theta )\))
and \(\{ X_a : 1 \le a \le 2n \}\) is an orthonormal (i.e. \(G_\theta (X_a , X_b ) = \delta _{ab}\)) local frame of \(H(M)\). Also \(\varPi _H : T(M) \rightarrow H(M)\) is the projection associated to the direct sum decomposition \(T(M) = H(M) \oplus {\mathbb {R}} T\). The divergence operator is meant with respect to the volume form \(\varPsi _\theta \), i.e.
for any \(C^1\) vector field \(X\) on \(M\) (here \({\mathcal L}_X\) is the Lie derivative in the direction \(X\)). \(\nabla ^H u \in C(H(M))\) is the horizontal gradient of \(u\). The differential operator \(\Delta _b\) is degenerate elliptic (in the sense of Bony [7]) yet subelliptic of order \(1/2\) (in the sense of Folland [20]) hence (by a result of Hörmander [22]) hypoelliptic (a property shared by elliptic operators). Therefore, the study of subelliptic harmonic maps fits into the larger program started by Jost and Xu [23], aimed to recovering known results on solutions to nonlinear elliptic systems of variational origin to the at least hypoelliptic case. The geometric interpretation in [4] is that \(\phi : (M, \theta ) \rightarrow (S, h)\) is subelliptic harmonic if and only if its vertical lift \(\phi \circ p : (C(M), F_\theta ) \rightarrow (S, h)\) is harmonic.
Let \(\phi : M \rightarrow S\) be a smooth map. If \(Y \in \mathfrak {X}(S)\) we set \(Y^\phi = Y \circ \phi \). Also for each \(X \in \mathfrak {X}(M)\) let \(\phi _*X \in C^\infty (\phi ^{-1} T(S))\) be given by \((\phi _*X)(x) = (d_x \phi ) X_x\) for any \(x \in M\). If \(\phi : M \rightarrow S\) is a CR map of strictly pseudoconvex CR manifolds and \(\theta \) and \(\Theta \) positively oriented contact structures on \(M\) and \(S\), there is a unique \(C^\infty \) function \(\lambda (\phi ) = \lambda (\phi ; \, \theta , \, \Theta ) : M \rightarrow (0, + \infty )\) such that \(\phi ^*\Theta = \lambda (\phi ) \, \theta \). Function \(\lambda (\phi )\) is the dilation of \(\phi \) as a map of the pseudohermitan manifolds \((M, \theta )\) and \((S, \Theta )\). We shall need the following
Lemma 1
Let \((M, T_{1,0}(M))\) and \((S, T_{1,0}(S))\) be two strictly pseudoconvex CR manifolds, of CR dimensions \(n\) and \(N = n + k\), \(k \ge 1\), respectively. Let \(\theta \) and \(\Theta \) be positively oriented contact forms on \(M\) and \(S\), respectively, and let \(T_\theta \in \mathfrak {X}(M)\) and \(T_\Theta \in \mathfrak {X}(S)\) be the corresponding Reeb vector fields. Then a CR map \(\phi : M \rightarrow S\) is subelliptic harmonic as a map of the pseudohermitian manifold \((M, \theta )\) into the Riemannian manifold \((S, g_\Theta )\) if and only if
and
for some \(c \in {\mathbb {R}}\). Here \(A_\Theta (V,W) = g_\Theta (V, \tau _\Theta W)\), \(V,W \in \mathfrak {X}(S)\), is the pseudohermitian torsion of \(\nabla ^\Theta \).
Proof
Let \(\nabla ^\theta \) and \(D^\theta \) be, respectively, the Tanaka-Webster and Levi-Civita connections of \((M, \theta )\) and \((M, g_\theta )\). Then (by identity (1.61) in [14], p. 37)
Here \(A_\theta (X,Y) = g_\theta (X , \tau _\theta Y)\), \(X,Y \in \mathfrak {X}(M)\) (the pseudohermitian torsion of \(\nabla ^\theta \)). Also \(\varOmega _\theta (X, Y) = g_\theta (X, J Y)\), i.e. \(\varOmega _\theta = - \mathrm{d}\theta \). By a result in [33] \(A_\theta \) is symmetric, i.e. \(A_\theta (X,Y) = A_\theta (Y, X)\) and traceless. Then (by (7) and \(J T_\theta = 0\))
for any \(X,Y \in C^\infty (H(M))\). Then (by (8) and \(\phi _*\circ J = J_S^\phi \circ \phi _*\))
hence
Here \(J_S^\phi = \phi ^{-1} J_S\) and \(G_\Theta ^\phi = \phi ^{-1} G_\Theta \) denote the pullbacks of \(J_S\) and \(G_\Theta \) (a complex structure and bundle metric in the pullback bundle \(\phi ^{-1} H(S) \rightarrow M\)), respectively. Let \(T_D\) denote the torsion tensor field of the linear connection \(D\). Since \(T_{D^\Theta } = 0\) and
(cf. identity (1.60) in [14], p. 37) the pseudohermitian second fundamental form \(\beta _b (\phi )\) is not symmetric yet
for any \(X,Y \in C^\infty (H(M))\). Therefore (by (9)–(10))
i.e.
Next a calculation based on (9) and (11) shows that
Again by the very definition of \(\beta _b (\phi )\) and
one has
and then (by \(\mathrm{Ker}(\Theta ) = H(S)\))
Finally substitution from (13) into (12) yields
Let \(\{ X_a : 1 \le a \le 2n \}\) be a local \(G_\theta \)-orthonormal frame of \(H(M)\), defined on the open set \(U \subset M\), such that \(X_{n+\alpha } = J X_\alpha \) for any \(1 \le \alpha \le n\). Then (by (14))
on \(U\). Taking into account the decomposition \(T(S) = H(S) \oplus {\mathbb {R}} T_\Theta \) it follows that \(\phi \) is subelliptic harmonic (as a map of \((M, \theta )\) into \((S, g_\Theta )\)) if and only if (5) holds and
let us apply \(J_S^\phi \) to (15) and use \(J_S^2 = - I + \Theta \otimes T_\Theta \) and \(\phi ^*\Theta = \lambda (\phi ) \, \theta \). We obtain
Finally (by \(T_\Theta \, \rfloor \, \mathrm{d}\Theta = 0\) and (16))
for any \(X \in C^\infty (H(M))\). Therefore, \(\lambda (\phi )\) is real valued and \(\overline{\partial }_b \lambda (\phi ) = 0\) hence \(\lambda (\phi ) = c\) for some \(c \in {\mathbb {R}}\). \(\square \)
Let \(\phi : S^{2n+1} \rightarrow S^{2N+1}\) be a CR map (\(N = n+k\), \(k \ge 0\)). The CR degree of \(\phi \) is defined by
where \(\omega _n = \mathrm{Vol} (S^{2n+1} , \, \varPsi _{\eta _0} ) = \int _{S^{2n+1}} \varPsi _{\eta _0}\) and \(\lambda (\phi ; \, \eta , \, E_0 )\) is the dilation of \(\phi \) with respect to the contact forms \(\eta \in {\mathcal P}_+\) and \(E_0\) on \(S^{2n+1}\) and \(S^{2N+1}\), respectively.
Lemma 2
Let \(\phi : S^{2n+1} \rightarrow S^{2N+1}\) be a CR map and \(1 \le p < \infty \). The \(L^p\) norm \(\Vert \lambda _0 (\phi ) \Vert _p\) of the dilation \(\lambda _0 (\phi )\) is a CR invariant if and only if \(p = n+1\). The CR degree of \(\phi \) is not a homotopy invariant.
Proof
If \(\eta = e^u \eta _0\) for some \(u \in C^\infty (S^{2n+1})\) then \(\varPsi _\eta = e^{(n+1)u} \varPsi _{\eta _0}\) and if \(\hat{\lambda }_0 (\phi ) = \lambda (\phi ; \, \eta , \, E_0 )\), i.e., \(\phi ^*E_0 = \hat{\lambda }_0 (\phi ) \, \eta \) then \(\hat{\lambda }_0 (\phi ) = e^{-u} \lambda _0 (\phi )\). Hence \(\Vert \lambda _0 (\phi ) \Vert _p = \Vert \hat{\lambda }_0 (\phi ) \Vert _p\) if and only if \(p = n+1\). To prove the second statement in Lemma 2 one exhibits an example. For instance, the 1-parameter family \(\{ \phi _t \}_{0 \le t \le 1} \subset C^\infty (S^3 , S^5 )\) in 4) of Theorem 1 shows that \(\phi _{{\mathbb {A}}_0}\) and \(\phi _{{\mathbb {A}}_1}\) are homotopic yet \(\mathrm{deg} (\phi _{{\mathbb {A}}_0} ) = 4\) and \(\mathrm{deg}(\phi _{{\mathbb {A}}_1}) = 7/3\) (explicit calculations of CR degrees are provided in Sect. 5). \(\square \)
Let us consider the de Rham cohomology class \(\alpha = (1/\omega _n ) \, \left[ \varPsi _{\eta _0} \right] \) so that \(\alpha \) is a generator of \(H^{2n+1}(S^{2n+1} , {\mathbb {R}}) \approx {\mathbb {R}}\) with \(\int _{S^{2n+1}} \alpha = 1\). If \(k = 0\) then the topological degree \(d_\phi \in {\mathbb Z}\) of \(\phi \) is given by \(\phi ^*\alpha = d_\phi \, \alpha \) where \(\phi ^*: H^{2n+1}(S^{2n+1}, {\mathbb {R}}) \rightarrow H^{2n+1}(S^{2n+1} , {\mathbb {R}})\) is the induced homomorphism of cohomology groups. Also
i.e., in the equidimensional case (17) is the ordinary degree of \(\phi \).
Theorem 1
For every \(\Phi \in \bigcup _{{\mathcal {C}} \in P^*(2,3)} {\mathcal {C}}\) let \(\phi : S^3 \rightarrow S^5\) be its boundary values. Then 1) both \(\phi _{{\mathbb {A}}_0} , \; \phi _{\mathbb {I}} : S^3 \rightarrow S^5\) are unstable subelliptic harmonic as maps of the pseudohermitian manifold \((S^3 , \eta _0 )\) into the Riemannian manifold \((S^5 , g_{E_0})\). 2) None of the maps \(\Phi \in {\mathbb {F}} \cup {\mathbb {A}}_1\) has subelliptic harmonic boundary values. 3) If \({\mathcal {C}} \in \{ {\mathbb {A}}_0 , \; {\mathbb {I}} \}\) then \(\{ U \circ \Phi _{\mathcal {C}} \circ u^{-1} : u \in \mathrm{U}(2), \;\; U \in \mathrm{U}(3) \}\) are all the proper holomorphic maps in \({\mathcal {C}}\) whose boundary values are subelliptic harmonic as maps of \((S^3 , \eta _0 )\) into \((S^5 , \, E_0 )\). 4) The 1-parameter family
is a deformation of \(\Phi _{{\mathbb {A}}_0}\) into \(\Phi _{{\mathbb {A}}_1}\) by proper holomorphic maps. If \(\phi _t : S^3 \rightarrow S^7\) is the boundary values of \(\Phi _t : {\mathbb {B}}_2 \rightarrow {\mathbb {B}}_4\) then \(\phi _t\) is a subelliptic harmonic map of \((S^3 , \eta _0 )\) into \((S^7 , \, j_3^*\left\{ \frac{i}{2} \left( \overline{\partial } - \partial \right) \sum _{A=1}^4 Z_A \overline{Z}_A \right\} )\) if and only if \(t = 0\). The CR degree of \(\phi _t\) is \(\mathrm{deg}(\phi _t ) = t^4 /3 - 2 t^2 + 4\) for every \(0 \le t \le 1\).
Statements 1) to 3) answer the question, posed by Minor et al. (cf. [9, 10]) whether one may distinguish among the elements of \(P^*(2, 3)\) based on the geometry of the second fundamental form of their representatives. Let \((Z^0 , Z^1 , Z^2 )\) be the Cartesian complex coordinates on \({\mathbb {C}}^3\). If \(\theta _0 = \frac{i}{2} \left( \overline{\partial } - \partial \right) (z \overline{z} + w \overline{w} )\) and \(\Theta _0 = \frac{i}{2} \left( \overline{\partial } - \partial \right) \sum _{A=1}^3 Z^A \overline{Z}_A\) with \(\overline{Z}_A = \overline{Z^A}\) then \(\mathrm{d} \theta _0 = i \left( \mathrm{d} z \wedge \mathrm{d}\overline{z} + \mathrm{d}w \wedge \mathrm{d} \overline{w} \right) \) and \(\mathrm{d}\Theta _0 = i \, d Z^A \wedge \mathrm{d}\overline{Z}_A\). The CR structures \(T_{1,0}(S^3 )\) and \(T_{1,0}(S^5 )\) are, respectively, the span of \({{\mathcal {Z}}} = \overline{w} \, \partial /\partial z - \overline{z} \, \partial /\partial w\) and \(T_j = \overline{Z}_3 \, \partial /\partial Z^j - \overline{Z}_j \, \partial /\partial Z^3\), \(j \in \{ 1,2 \}\). Consequently the Levi forms of \(S^3\) and \(S^5\) are \(G_{\theta _0} (Z , \overline{Z}) = 1/2\) and
The Reeb vector fields \(T_{\eta _0} \in \mathfrak {X}(S^3 )\) and \(T_{E_0} \in \mathfrak {X}(S^5 )\) associated to the contact forms \(\eta _0\) and \(E_0\) are given by \((d j_1 ) T_{\eta _0} = i \left( z \, \partial /\partial z + w \, \partial /\partial w \right) +\) complex conjugate and \((d j_2 ) T_{E_0} = i \, Z^A \, \partial /\partial Z^A +\) complex conjugate. Let \(\eta _0 = j_1^*\theta _0\) and \(E_0 = j_2^*\Theta _0\). If \(\lambda _0 (\phi ) = \lambda (\phi ; \, \theta _0 , \, \Theta _0 )\) is the dilation of \(\phi \) as a map of \((S^3 , \, \eta _0 )\) and \((S^5 , \, E_0 )\) then a straightforward calculation shows that
The standard sphere \((S^5 , \, E_0 )\) has vanishing pseudohermitian torsion (\(A_{E_0} = 0\)). Then (by Lemma 1) \(\phi _{{\mathbb {A}}_0}\) and \(\phi _{\mathbb {I}}\) are subelliptic harmonic as maps of \((S^3 , \, \eta _0 )\) into \((S^5 , \, g_{E_0})\). The proof of the stability statement is relegated to Sect. 4. To prove 2) in Theorem 1 we recall that \(\mathrm{Hol}({\mathbb {B}}_N ) = \{ U \, \varphi _A : U \in \mathrm{U}(N), \;\; A \in {\mathbb {B}}_N \}\) where
For each \(\zeta \in \mathrm{Hol}({\mathbb {B}}_N )\) let \(\zeta _S \in \mathrm{Aut}_\mathrm{CR} (S^{2N-1})\) be the boundary values of \(\zeta \). If \(\xi \in \mathrm{Hol}({\mathbb {B}}_2 )\) and \(\zeta \in \mathrm{Hol}({\mathbb {B}}_3 )\) we need to establish
Lemma 3
Let \({\mathcal {C}} \in P^*(2 , 3)\) and \(\Phi \in {\mathcal {C}}\). Let \(\phi \in {\mathcal {C}}_b\) be the boundary values of \(\Phi \). If \(\hat{\eta }_0 = \xi _S^*\eta _0\) and \(\hat{E}_0 = \zeta _S^*E_0\) then the following statements are equivalent
-
(i) The boundary values \(\hat{\phi }\) of \(\hat{\Phi } = \zeta \circ \Phi \circ \xi ^{-1}\) is a subelliptic harmonic map of \((S^3 , \, \eta _0 )\) into \((S^5 , \, g_{E_0})\).
-
(ii) \(\phi \) is a subelliptic harmonic map of \((S^3 , \, \hat{\eta }_0 )\) into \((S^5 , \, g_{\hat{E}_0} )\).
Proof
Assume (i). Then (by Lemma 1) \(\lambda _0 (\hat{\phi } ) = \lambda (\hat{\phi } ; \, \eta _0 , \, E_0 )\) is a constant \(c \in {\mathbb {R}}\) and \(\lambda _0 (\hat{\phi }) \, T_{E_0}^{\hat{\phi }} - \hat{\phi }_*\, T_{\eta _0} = 0\). Note that \(\hat{\phi } = \zeta _S \circ \phi \circ \xi _S^{-1}\). Then for every \(q \in S^3\)
where \(\hat{\eta }_0 = \xi _S^*\eta _0\). Identity (18) follows from \(\square \)
Lemma 4
Let \(\eta _0 = j_n^*\theta _0\) be the canonical contact form on \(S^{2n+1}\) where \(\theta _0 = \frac{i}{2} \sum _{k=1}^{n+1} z^k \, \mathrm{d} \overline{z}_k +\) complex conjugate. If \(\xi \in \mathrm{Hol} ({\mathbb {B}}_{n+1})\) and \(\hat{\eta }_0 = \left( \xi _S \right) ^*\eta _0\) then the Reeb vector fields of \(\eta _0\) and \(\hat{\eta }_0\) are related by
We set \(V = \left( (\xi _S^{-1} )_*\, T_{\eta _0} \right) \circ \xi _S\) so that the claim of Lemma 4 becomes \(V = T_{\hat{\eta }_0}\). It suffices to check that
as \(T_{\hat{\eta }_0} \in \mathfrak {X}(S^{2n+1})\) is determined by these properties (cf. Proposition 1.2 in [14], p. 8). The first identity in (20) follows from \(\eta _0 \left( T_{\eta _0} \right) = 1\). As to the second identity, for every \(q \in S^{2n+1}\) and \(X \in \mathfrak {X}(S^{2n+1})\)
where \(Y \in \mathfrak {X}(S^{2n+1})\) is defined by \(Y = \left( (\xi _S )_*X \right) \circ \xi _S^{-1}\). Let us go back to the proof of Lemma 3. By (18)
Also (by Lemma 4 with \(n=2\))
hence (by applying \(d_{\zeta _S (\phi (\xi _S^{-1} (q)))} \zeta _S^{-1}\) to (21))
On the other hand the identities
yield
and then \(\lambda (\phi ; \, \hat{\eta }_0 , \, \hat{E}_0 ) = c\). Together with (22) and Lemma 1 this implies (ii) in Lemma 3. The proof of the opposite implication is similar.
Let \({\mathcal {C}} \in P^*(2,3)\) and \(\Phi \in {\mathcal {C}}\). Let \(\phi \in {\mathcal {C}}_b\) be the boundary values of \(\Phi \). The problem set forth is to look for \(\xi \in \mathrm{Hol}({\mathbb {B}}_2 )\) and \(\zeta \in \mathrm{Hol}({\mathbb {B}}_3 )\) such that the boundary values \(\hat{\phi }\) of \(\hat{\Phi } = \zeta \circ \Phi \circ \xi ^{-1}\) be subelliptic harmonic as a map of the pseudohermitian manifold \((S^3 , \, \eta _0 )\) into the Riemannian manifold \((S^5 , \, g_{E_0} )\). Equivalently (by Lemmas 1 and 3) if \(\hat{\eta }_0 = e^{2f} \eta _0\) and \(\hat{E}_0 = e^{2 F} E_0\) for some \(f \in C^\infty (S^3 )\) and \(F \in C^\infty (S^5 )\) then one should determine \(\xi \) and \(\zeta \) such that
for some \(c \in {\mathbb {R}}\). We need
Lemma 5
-
(i) The canonical contact form on \(S^{2N-1}\) is \(\mathrm{U}(N)\)-invariant, i.e., \(U^*\Theta _0 = \Theta _0\). (ii) Let \(A \in {\mathbb {B}}_N\). If \(\hat{E}_0 = \left( \varphi _A \right) _S^*\, E_0\) then \(\hat{E}_0 = e^{2 F} \, E_0\) where \(F \in C^\infty (S^{2N-1})\) is given by
$$\begin{aligned} F(Z) = \log \frac{s_A}{\left| 1 - \langle Z , A \rangle \right| } , \;\;\; Z \in S^{2N-1}. \end{aligned}$$(26)In particular the CR automorphism \(\left( \varphi _A \right) _S \in \mathrm{Aut}_\mathrm{CR} (S^{2N-1})\) is isopseudohermitian (i.e., \(\varphi _A^*\Theta _0 = \Theta _0\)) if and only if \(\varphi _A\) is the identity.
Proof
(i) Let \(U = \left[ U^j_k \right] \in \mathrm{U}(N)\). Then \(U \, \mathrm{d} Z^j = U^j_k \, \mathrm{d} Z^k\). Finally as \(\Theta _0 = - \frac{i}{2} \left( \overline{Z}_j \, d Z^j \right) +\) complex conjugate and \(Z^j \circ U = U^j_k Z^k\), one has
(ii) If \(\lambda _0 ( (\varphi _A )_S ) = \lambda ((\varphi _A )_S ; \, E_0 , \, E_0 )\) is the dilation of \((\varphi _A )_S\) as a CR map of \((S^{2N-1} , \, E_0 )\) in itself, then
yielding (26). In particular \(\lambda _0 ((\varphi _A )_S ) \equiv 1\) if and only if \(A = 0\). \(\square \)
Lemma 6
Let \(a \in {\mathbb {B}}_2\) and \(A \in {\mathbb {B}}_3\) and let us consider the automorphisms \(\xi = u \, \varphi _a\) and \(\zeta = U \, \varphi _A\) with \(u \in \mathrm{U}(2)\) and \(U \in \mathrm{U}(3)\). Then \(\hat{\phi } = \zeta _S \circ \phi \circ \xi _S^{-1}\) is subelliptic harmonic as a map of \((S^3 , \, \eta _0 )\) into \((S^5 , \, g_{E_0})\) if and only if
for some \(c \in {\mathbb {R}}\) and
where \(F^j = \left( G_{\Theta _0} \right) ^{j \overline{k}} T_{\overline{k}} (F)\) and \(f^1 = \left( G_{\theta _0} \right) ^{1 \overline{1}} \overline{{\mathcal {Z}}} (f)\) (and \(F^{\overline{j}} = \overline{F^j}\) and \(f^{\overline{1}} = \overline{f^1}\)). Moreover
Proof
Let us use Lemma 5 for \(N \in \{ 2,3 \}\). Then substitution from (26) into (24) yields (28). As \(\hat{\eta }_0 = e^{2f} \eta _0\) one has (by the proof of Lemma 2.6 in [14], p. 136)
Here \(J_{S^3}\) denotes the complex structure along \(H(S^3)\) and \(\left( \nabla ^{\eta _0} f \right) ^H\) is the horizontal gradient of \(f\) (as a scalar function on the pseudohermitian manifold \((S^3 , \, \eta _0 )\)). As \(\left( G_{\eta _0} \right) _{1 \overline{1}} = \frac{1}{2}\) and (by Lemma 5)
one has
hence the horizontal gradient of \(f\) reads \(\left( \nabla ^{\eta _0} \right) ^H f = f^1 \, {{\mathcal {Z}}} + f^{\overline{1}} \, \overline{{\mathcal {Z}}}\) with \(f^1 = 2 \, \overline{{\mathcal {Z}}}(f)\) given by (31). Similar to (32)
and \(\left( \nabla ^{E_0} \right) ^H F = F^j T_j + F^{\overline{j}} F_{\overline{j}}\) with \(F^j\) given by (30). Substitution from (32)–(33) into (25) yields (29). \(\square \)
Lemma 7
If \(\Phi = \Phi _{\mathbb {F}}\) then Eqs. (28)–(29) are equivalent to
on \(z \overline{z} + w \overline{w} = 1\), where \(k_{a,A} = \left( s_A /s_a \right) ^2\). Consequently \(\hat{\phi }_{\mathbb {F}} : (S^3 , \, \eta _0 ) \rightarrow (S^5 , \, g_{E_0})\) is not subelliptic harmonic.
Proof
Note that
Proof of Lemma 7 is by contradiction. Assume \(\hat{\phi }_{\mathbb {F}}\) is subelliptic harmonic for some \(a \in {\mathbb {B}}_2\) and \(A \in {\mathbb {B}}_3\). Then identities (35)–(36) at \((0,1)\) are
Next identity (34) at \((z,w) \in \{ \pm (1,0), \, \pm (0,1), \, (i, 0), \, (0, i) \}\) yields
and other five identities of the kind. Subtraction in pairs leads to \(a_1 = 0\). Thus (by (37)) \(A_2 = 0\). Similarly
Let us evaluate (34) at \((z,w) \in \{ \pm (1/\sqrt{2} , \, 1 /\sqrt{2}), \; \pm (1 /\sqrt{2} , \, - 1/\sqrt{2} ) \}\) to derive (as \(\lambda _0 (\phi _{\mathbb {F}})_{(z,w)} = 9/4\))
and other three equations of the kind. Subtraction in pairs furnishes
Similarly we may evaluate (34) at
so that to obtain (after some unenlightening calculations that we omit)
Finally identities (41)–(42) give
A comparison among (39) and (43) shows that \(A_3 = 0\) and then \(a_2 = 0\). We may conclude that \(a = 0\) and \(A = 0\), i.e., \(\hat{\phi }_{\mathbb {F}} = \phi _{\mathbb {F}}\), a contradiction. \(\square \)
Lemma 8
If \(\Phi = \Phi _{{\mathbb {A}}_1}\) then Eqs. (28)–(29) are equivalent to
on \(|z|^2 + |w|^2 = 1\). Consequently \(\hat{\phi }_{{\mathbb {A}}_1} : (S^3 , \, \eta _0 ) \rightarrow (S^5 , \, E_0 )\) is not subelliptic harmonic.
Proof
Assume \(\hat{\phi }_{{\mathbb {A}}_1}\) to be subelliptic harmonic. Evaluation of (45)–(46) at \((z,w) \in \{ \pm (0,1), \; \pm (0, i) \}\) gives (as \(\lambda _0 (\phi _{{\mathbb {A}}_1} )_{(z,w)} = 2\)) \(A = (0, \, 0, \, A_3 )\) and \(a = (0, \, a_2 )\) and Eq. (44) may be written
Identity (47) at \((1,0)\) yields \(k_{a,A} = c\) and evaluation of (47) at \((z,w) \in \{ \pm (0,1), \; \pm (0, i) \}\) furnishes
hence \(a_2 = 0\), i.e., \(a = 0\). Identities \(\left| 1 \pm A_3 \right| ^2 = 2\) then imply \(A_3 \in \{ \pm i \}\), i.e., \(A \in S^5\), a contradiction. \(\square \)
Lemma 9
If \(\Phi = \Phi _{\mathbb {I}}\) then Eqs. (28)–(29) are equivalent to
Consequently (i) \(\varphi _A \circ \phi _{\mathbb {I}} \circ \varphi _a^{-1} : (S^3 , \, \eta _0 ) \rightarrow (S^5 , \, E_0 )\) is subelliptic harmonic if and only if \(A = (0, a)\). In particular ii)
are all the subelliptic harmonic maps \((S^3 , \, \eta _0 ) \rightarrow (S^5 , \, E_0 )\) in the boundary values \({\mathbb {I}}_b\) of the spherical equivalence class \({\mathbb {I}} \in P^*(2,3)\). iii) Every map in (50) is unstable.
Proof
Since \(\lambda _0 (\phi _{\mathbb {I}}) = 1\) and \(\left( \phi _{\mathbb {I}} \right) _*{{\mathcal {Z}}} = T_2^{\phi _{\mathbb {I}}}\) Eqs. (28)–(29) are equivalent to (48)–(49). Next [by (30)–(31)] equations (49) may be written
Setting \((z,w) \in \{ \pm (0, 1), \; \pm (0,i) \}\) gives either \(a_1 = 0\), and then \(A_2 = 0\), or \(A_3 = a_2\) and then \(A_2 = a_1\). So far either \(a = (0, a_2 )\) and \(A = (0,0,A_3 )\) or \(a_1 \ne 0\) and \(A = (0,a)\). Next we set \((z,w) \in \{ \pm (0,1), \; \pm (0, i) \}\) in (48) and obtain
together with other three equations of the kind. These imply
Consequently \(A = (0, a)\) and, conversely, if \(A = (0,a)\) then the (second) relation (51) is identically satisfied. To prove statement (ii) in Lemma 9 one may observe that when \(A = (0, a)\) and \(q^\prime = \varphi _a (q)\) one has \(L_A (\Phi _{\mathbb {I}} (q^\prime )) = (0, \, L_a (q^\prime ))\) hence \(\varphi _A \circ \Phi _{\mathbb {I}} \circ \varphi _a = \Phi _{\mathbb {I}}\) (by \(\varphi _a^2 = 1_{{\mathbb {B}}_2}\)). Statement (iii) is proved in Sect. 4. \(\square \)
Lemma 10
If \(\Phi = \Phi _{{\mathbb {A}}_0}\) then Eqs. (28)–(29) are equivalent to
Consequently (i) the boundary values of \(\varphi _A \circ \Phi _{{\mathbb {A}}_0} \circ \varphi _a\) is a subelliptic harmonic map of \((S^3 , \eta _0 )\) into \((S^5 , g_{E_0} )\) if and only if \(A = 0\) and \(a = 0\). In particular (ii)
are all the subelliptic harmonic maps in the boundary values \(\left( {\mathbb {A}}_0 \right) _b\) of the spherical equivalence class \({\mathbb {A}}_0 \in P^*(2,3)\). (iii) All maps in (55) are unstable.
Proof is similar to that of Lemma 9, and thus omitted. Lemmas 7 to 10 imply 1) to 3) in Theorem 1, except for the instability statements. A calculation shows that \(\lambda _0 (\phi _t )_{(z,w)} = 2 - t^2 |z|^2\), for every \((z,w) \in S^3\) and \(0 \le t \le 1\), hence (by Lemma 1) \(\phi _t\) subelliptic harmonic if and only if \(t = 0\).
4 Stability theory
Let \(\theta \in {\mathcal P}_+\) be a positively oriented contact form on the sphere \(S^3 \subset {\mathbb {C}}^2\), thought of as a strictly pseudoconvex CR manifold with the CR structure induced by the complex structure of \({\mathbb {C}}^2\). Let \(\phi \in C^\infty (S^3 , \, S)\) be a subelliptic harmonic map of \((S^3 , \, \theta )\) into the Riemannian manifold \((S, h)\). Let \(\left\{ \phi _{s,t} \right\} _{- \epsilon < s,t < \epsilon }\) be a smooth 2-parameter variation of \(\phi \), i.e., \(\phi _{0,0} = \phi \). Let
and let \(V,W \in C^\infty (\phi ^{-1} T(S^{3}))\) be defined by
If \(X \in \mathfrak {X}(S^{3})\) let \(\tilde{X}\) be the tangent vector field on \(S^{3} \times (- \epsilon , \epsilon )^2\) given by
Let \(U = \{ (z,w) \in S^3 : w \ne 0 \}\) and let us set \(X_1 = {{\mathcal {Z}}} + \overline{{\mathcal {Z}}}\) and \(X_2 = i \left( {{\mathcal {Z}}} - \overline{{\mathcal {Z}}} \right) \) (so that \(\{ X_1 , \, X_2 \} \subset C^\infty (U, H(S^3 ))\) is a local \(G_{\eta _0}\)-orthonormal frame). Note that
for every \(x \in U\) and \(- \epsilon < s,t < \epsilon \). Let \(f : (-\epsilon , \epsilon )^2 \rightarrow {\mathbb {R}}\) be given by \(f(s,t) = E \left( \phi _{s,t} \right) \) for every \((s,t) \in (-\epsilon , \epsilon )^2\). Only the local expression (56) is needed in order to compute \(\partial f/\partial t\). One has
\(\Big (\)as \(\left[ \partial /\partial t , \, \tilde{X} \right] = 0\) for every \(X \in \mathfrak {X}(S^{3})\Big )\)
(as \(\left( \nabla ^S \right) ^\Phi h^\Phi = 0\))
Let us fix \(s, t \in (- \epsilon , \epsilon )\) and consider \(X_{s,t} \in C^\infty (H(S^{3}))\) determined by
for every \(Y \in C^\infty (H(S^{3}))\). Then
(as \(\nabla G_{\eta _0} = 0\))
so that (by Green’s lemma)
where \(\beta (\Phi ) (X,Y) \in C^\infty \left( \Phi ^{-1} T (S) \right) \) is given by
for any \(X,Y \in \mathfrak {X}(S^{3})\). The local expression of the integrand in (57) again suffices to compute \(\{ \partial ^2 f/\partial s \partial t \}_{s=t=0}\). One has
and
hence
at \((x,0,0)\) for every \(x \in U\). To recognize a divergence term one needs to switch covariant derivatives
hence curvature \(R^\Phi \equiv R_{(\nabla ^S )^\Phi }\) (of the connection \((\nabla ^S )^\Phi \) in the vector bundle \(\Phi ^{-1} T(S) \rightarrow S^{3} \times (- \epsilon , \epsilon )^2\)) comes in. Right hand side of (58) becomes
or (by \(\left[ \partial /\partial s , \, \tilde{X}_a \right] = 0\) and \(\left( \nabla ^S \right) ^\Phi h^\Phi = 0\))
(evaluated at \((x,0,0)\)). Let us consider the vector field \(X_\phi \in C^\infty (H(S^{3}))\) determined by
for every \(Y \in C^\infty (H(S^{3}))\). Then
hence (59) becomes
or
so that
which is the second variation formula (3) in Sect. 1. Here \((( \nabla ^S )^\phi )^H V\) is the restriction of \(( \nabla ^S )^\phi V\) to \(H(S^{3})\), providing a first-order differential operator
Also \((R^S )^\phi \equiv \phi ^{-1} R^S\) is the pullback by \(\phi \) of the curvature tensor field \(R^S\) of \(\nabla ^S\). It is noteworthy that derivation of (60) also relied on the symmetries of the Riemann–Christoffel 4-tensor associated to \(R^S\). The pointwise inner product \((h^\phi )^*\) in (60) is naturally induced by \(h^\phi \), i.e., locally
for any \(\varphi , \psi \in C^\infty (U, H(S^{3})^*\otimes \phi ^{-1} T(S))\) and any \(x \in U\). We shall also need the \(L^2\) inner products
on \(C^\infty (\phi ^{-1} T(S))\) and \(C^\infty (H(S^{3})^*\otimes \phi ^{-1} T(S))\), respectively. To simplify writing we adopt the notations
Let
be the formal adjoint of \(d_\phi ^H\), i.e.,
for every \(V \in C^\infty (\phi ^{-1} T(S))\). We need
Lemma 11
For every \(\varphi \in C^\infty (H(S^{3})^*\otimes \phi ^{-1} T(S))\)
where the covariant derivative \(D^\phi \varphi \) is given by
for any \(X \in \mathfrak {X}(S^{3})\) and any \(Y \in C^\infty (H(S^{3}))\).
Proof
Locally
Let us define the vector field \(X_\varphi \in C^\infty (H(S^{3}))\) by setting
Then
hence
Integration over \(S^{3}\) in (62) and Green’s lemma yield (61). \(\square \)
We shall need the second-order differential operator
referred to as the rough sublaplacian. Lemma 11 for \(\varphi = d_\phi ^H V\) shows that locally
on \(U\) for every \(V \in C^\infty (\phi ^{-1} T(S))\). The Jacobi operator is
hence first variation formula (60) reads
Let \({\mathfrak {shar}} (S^{3} , \, S)\) denote the set of all subelliptic harmonic maps of \((S^{3}, \, \theta )\) into \((S, h)\). The Hessian of \(E\) at \(\phi \in {\mathfrak {shar}}(S^{3} , \, S)\) is
for any \(V,W \in C^\infty (\phi ^{-1} T(S))\). As a consequence of (64) if \(V \in C^\infty (\phi ^{-1} T(S))\) and \(\{ \phi _t \}_{|t| < \epsilon } \subset C^\infty (S^{3} , S)\) is a smooth 1-parameter variation of \(\phi \) inducing \(V\) then
(apply (64) with \(\phi _{s,t} = \phi _{s+t}\)).
To build a stability theory based on \(\mathrm{Hess}_b (E)_\phi \), we introduce the following notions. For every \(\phi \in {\mathfrak {shar}} (S^{3} , S)\) let \(\mathrm{null}_b (\phi )\) denote the dimension over \(\mathbb R\) of the nullity space
Moreover, let \(\mathrm{ind}_b (\phi )\) be the least upper bound of the set of all dimensions \(\dim _{\mathbb {R}} S\) of all subspaces \(S \subset C^\infty (\phi ^{-1} T(S))\) on which the Hessian of \(E\) at \(\phi \) is negative definite, i.e., \(\mathrm{Hess}_b (E)_\phi (V,V) < 0\) for every \(V \in S\).
A subelliptic harmonic map \(\phi \in {\mathfrak {shar}} (S^{3}, S)\) is (weakly) stable if \(\mathrm{ind}_b (\phi ) = 0\), i.e., \(\mathrm{Hess}_b (E)_\phi (V,V) \ge 0\) for every \(V \in C^\infty (\phi ^{-1} T(S))\) (equivalently \(\{ \mathrm{d}^2 E(\phi _t )/\mathrm{d}t^2 \}_{t=0} \ge 0\) for every smooth 1-parameter variation \(\{ \phi _t \}_{|t| < \epsilon }\) of \(\phi \), cf. (65)). Also \(\phi \in {\mathfrak {shar}}(S^{3} , \, S)\) is unstable if \(\mathrm{ind}_b (\phi ) > 0\).
As an example, we look at constant maps, i.e., \(\phi (x) = y_0\) with \(y_0 \in S\), for any \(x \in S^{3}\). Each fiber in the pullback bundle \(\phi ^{-1} T(S) \rightarrow S^{3}\) is then a copy of \(T_{y_0} (S)\). Consequently if \(\{ v_i : 1 \le i \le m \} \subset T_{y_0} (S)\) is a fixed linear basis then
is a global frame in \(\phi ^{-1} T(S)\) hence each \(V \in C^\infty (\phi ^{-1} T(S))\) may be represented as \(V = \sum _{i=1}^m \varphi _i V_i\) for some \(\varphi _i \in C^\infty (S^{3})\). Also
so that
where \(\Delta _b\) is the sublaplacian of \((S^{3} , \, \theta )\). We may conclude that
Proposition 1
If \(\phi : S^{3} \rightarrow S\) is the constant map \(\phi (x) = y_0\) then
-
(i) the spectrum \(\sigma (J_b^\phi )\) of the Jacobi operator is the spectrum \(\sigma (\Delta _b )\) of the sublaplacian \(\Delta _b\) counted \(m = \dim (S)\) times. In particular \(\sigma (J_b^\phi )\) doesn’t depend on \(y_0 \in S\).
-
(ii) \(\mathrm{ind}_b (\phi ) = 0\), i.e., \(\phi \) is stable and \(\mathrm{null}_b (\phi ) = m\).
Proof
Statement (i) follows from (68). Let \(\{ v_i : 1 \le i \le m \} \subset T_{y_0} (S)\) be an orthonormal basis, i.e., \(h_{y_0} (v_i , \, v_j ) = \delta _{ij}\). Then (by Green’s lemma)
Let \(V \in C^\infty (\phi ^{-1} T(S))\) such that \(\mathrm{Hess}_b (E)_\phi (V, \, \cdot \, ) = 0\). Then (by (69)) \(\overline{\partial }_b \varphi _i = 0\) for any \(1 \le i \le m\) that is \(\varphi _i\) are real-valued CR functions on \(S^{3}\). Hence \(\varphi _i (x) = c_i\) for some \(c_i \in {\mathbb {R}}\) and then the nullity space of \(\mathrm{Hess}_b (E)\) is isomorphic to \({\mathbb {R}}^m\). \(\square \)
Let \(({\mathbb T}^m , h_0 )\) be a flat torus and \(\phi \in {\mathfrak {shar}}(S^{3} , {\mathbb T}^m )\) a subelliptic harmonic map. Subelliptic harmonic maps with values in a Riemannian manifold of nonpositive sectional curvature are stable (cf. [4], p. 735) hence already \(\mathrm{ind}_b (\phi ) = 0\). Let \(\{ Y_i : 1 \le i \le m \} \subset \mathfrak {X}({\mathbb T}^m )\) be parallel and linearly independent at each \(y \in {\mathbb T}^m\). If \(V_i = Y_i \circ \phi \) then (66)–(67), and therefore (68), will still hold hence \(\sigma (J_b^\phi )\) for every \(\phi \in {\mathfrak {shar}}(S^{3} , {\mathbb T}^m )\) coincides with the spectrum \(\sigma (J_b^{\phi _0} )\) of any constant map \(\phi _0 : S^{3} \rightarrow {\mathbb T}^m\). Thus, in general, the spectrum of the Jacobi operator \(J_b^\phi \) doesn’t determine the subelliptic harmonic map \(\phi \).
As another example, let \(\phi = 1_{S^{3}}\) be the identity map, i.e., \(\phi (x) = x\) for any \(x \in S^{3}\).
Proposition 2
\(1_{S^{3}} \in {\mathfrak {shar}}(S^{3} , \, S^{3})\) as a map of \((S^{3} , \, \theta )\) into \((S^{3} , \, g_\theta )\).
Proof
Let \((x^i )\) be the stereographic coordinates on \(U {\setminus } \{ (0,i) \}\). Then \(\phi ^i = x^i\) and the identity
yields
on \(U\) where \(b_a^i \in C^\infty (U)\) are given by \(X_a = b_a^i \, \partial /\partial x^i\). Also \((\varGamma ^\nabla )^i_{jk}\) are the local coefficients of the Tanaka-Webster connection \(\nabla \) of \((S^{3} , \, \theta )\) with respect to \((U, x^i )\). If \(S = S^{3}\) and \(h = g_{\eta _0}\) then the Levi-Civita connection \(\nabla ^S\) is related to \(\nabla \) by (cf. [14])
where \(\varOmega _0 = - \mathrm{d}\eta _0\). The local coordinate version of (71) is
where \(\partial _j = \partial /\partial x^j\). Let us contract with \(b_a^j \, b_a^k\) and sum over \(1 \le a \le 2\). It follows that
hence (by (70)) the functions \(\phi ^i\) satisfy the subelliptic harmonic map system (4). \(\square \)
Proving a pseudohermitian analog to Smith’s theorem (cf. [26]) for \(\phi = 1_{S^{3}}\) is an open problem. The stability statements in Lemmas 9–10 and Theorem 1 follow from
Lemma 12
Every subelliptic harmonic map \(\phi : S^3 \rightarrow S^5\) of \((S^3 , \, \eta _0 )\) into \((S^5 , \, g_{E_0} )\) is unstable.
Proof
Let \(\mathrm{tan}_Z : T_Z ({\mathbb {C}}^3 ) \rightarrow T_Z (S^5 )\) and \(\mathrm{nor}_Z : T_Z ({\mathbb {C}}^3 ) \rightarrow \nu (j_2 )\) be the projections associated to the direct sum decomposition
where \(\nu (j_2 )\) is the normal bundle of the immersion \(j_2 : S^5 \rightarrow {\mathbb {C}}^3\) (a real line bundle over \(S^5\)). Let \(\phi \in {\mathfrak {shar}}(S^3 , S^5 )\). If \((Y^j )\) are Cartesian coordinates on \({\mathbb {R}}^6\) then let \(V_k = \left[ \mathrm{tan} \left( \partial /\partial Y^k \right) ^{j_2} \right] ^\phi \). Let \(N\) be the outward unit normal of \({\mathbb {B}}_3\). Let \(W_\xi \) be the Weingarten, or shape, operator of the immersion \(j_2\), associated to the normal section \(\xi \in C^\infty (\nu (j_2 ))\), and let \(W_\xi ^\phi \) be the pullback of \(W_\xi \) (an endomorphism of the pullback bundle \(\phi ^{-1} T(S^5 ) \rightarrow S^3\)). Then (by the Gauss and Weingarten formulas, cf. e.g., [8])
Let \(U_+ = \{ Z \in S^5 : \mathrm{Im}(Z_3 ) > 0 \}\) and \(\chi : U_+ \rightarrow {\mathbb D}\) the projection \(\chi (Z) = \left( Z^\prime , \, \mathrm{Re} (Z_3 ) \right) \) with \(Z = (Z_1 , Z_2 )\) (a local chart on \(S^5\)). Here \({\mathbb D} = \{ \xi \in {\mathbb {R}}^5 : |\xi | < 1 \}\). Let \((\xi ^\alpha )\) be the corresponding local coordinates on \(U_+\). Then
Since \(j_2\) is totally umbilical \(W_N X = - X\) so that
where \(\phi ^\alpha = \xi ^\alpha \circ \phi \) and \(|\phi |^2 = \sum _{\alpha = 1}^5 \phi _\alpha ^2\) (with \(\phi _\alpha = \phi ^\alpha \)). Thus
with \(h = g_{E_0}\). By the Gauss equation (for \(j_2\), cf. [8])
one has
or (by taking the inner product with \(V_k\) and summing over \(1 \le k \le 5\))
hence \(\mathrm{ind}_b (\phi ) > 0\). \(\square \)
5 CR degree
The scope of this section is to establish the following
Theorem 2
Let \(\phi : S^{2n+1} \rightarrow S^{2N+1}\) be a CR map. Then (i)
(ii) For every \({\mathcal {C}} \in P^*(2,3)\) there is a constant \(0 < \varLambda _{\mathcal {C}} \le 1\) such that \(\mathrm{deg}(\phi _{\mathcal {C}} ) = \varLambda _{\mathcal {C}} \, p(\Phi _{\mathcal {C}} )^2\) with \(\varLambda _{\mathcal {C}} = 1\) if \(\phi _{\mathcal {C}}\) is subelliptic harmonic and \(\varLambda _{\mathcal {C}} < 1\) otherwise.
Also (iii) for every \({\mathcal {C}} \in P^*(2, 3)\) and every \(\phi \in {\mathcal {C}}_b\) one has \(\mathrm{deg}(\phi ) \approx \mathrm{deg}(\phi _{\mathcal {C}})\), i.e.,
where \(\Phi = \zeta \circ \Phi _{\mathcal {C}} \circ \xi ^{-1}\), \(\zeta = U \circ \varphi _A\) and \(\xi = u \circ \varphi _a\) with \(A \in {\mathbb {B}}_3\), \(a \in {\mathbb {B}}_2\), \(U \in \mathrm{U}(3)\) and \(u \in \mathrm{U}(2)\).
Here \(\Vert f \Vert _\infty = \sup _{x \in S^{2n+1}} |f(x)|\) for every \(f \in C(S^{2n+1}, {\mathbb {R}})\). Also, given a polynomial holomorphic map \(\Phi = \left( \Phi _1 , \ldots , \Phi _N \right) : {\mathbb {B}}_n \rightarrow {\mathbb {B}}_N\) we denote by \(p(\Phi ) \in {\mathbb N}\) the maximum of the degrees of the polynomials \(\Phi _j\) (\(1 \le j \le N\)). Statement (ii) in Theorem 2 follows from Lemma 1. For the calculation of the CR degrees of \(\phi _{\mathcal {C}}\) with \({\mathcal {C}} \in \{ {\mathbb {F}}, \; {\mathbb {A}}_1 \}\) let \(U_\pm = \{ (z,w) \in S^3 : \pm \mathrm{Im}(w) > 0 \}\) and let \(\psi _\pm : {\mathbb D} \rightarrow U_\pm \) be the parameterizations \(\psi _\pm (\xi ) = (\xi , \, \pm \sqrt{1 - |\xi |^2} )\) where \({\mathbb D} = \{ \xi \in {\mathbb {R}}^3 : |\xi | < 1 \}\) is the unit disk. Then
Next, as \(\psi _-\) preserves (respectively, \(\psi _+\) reverses) the orientation
hence
where \(p(\Phi _{\mathbb {F}}) = 3\) and \(\xi ^\prime = (\xi _1 , \, \xi _2 )\). In spherical coordinates \((\rho , \varphi , \theta )\) with \(0 \le \rho < 1\), \(0 \le \varphi \le \pi \) and \(0 \le \theta \le 2\pi \) the CR degree of \(\phi _{\mathbb {F}}\) is \(8 \pi \, p(\Phi _{\mathbb {F}})/\omega _1\) times
If \(I_m = \int _0^\pi \sin ^m \varphi \; \mathrm{d}\varphi \) then \((2n+1) I_{2n+1} = 2n \, I_{2n-1}\) for every \(n \ge 1\). Hence
so that
where \(I^m = \int _0^{\pi /2} \sin ^m t \; \mathrm{d}t\). Also \(2n I^{2n} = (2n-1) I^{2n-2}\) for every \(n \ge 1\) hence
so that \(J = 7 \pi /20\) and \(\mathrm{deg}(\phi _{\mathbb {F}}) = (14 \pi ^2 /5 \omega _1) \, p\left( \Phi _{\mathbb {F}} \right) ^2\). Moreover
and one may conclude
with \(\varLambda _{\mathbb {F}} = 7/10\). Similarly
where \(\varLambda _{{\mathbb {A}}_1} = 7/12\). To prove (75) one notes that (by Lemma 5)
For every \((z,w) \in S^3\)
hence
Moreover
as \((u^{-1} )^*\varPsi _{\eta _0} = \varPsi _{\eta _0}\) and \(\varphi _a^{-1} = \varphi _a\). Note that (cf. Lemma 5)
so that, again by a change of variables under the integral sign
Finally \(|1 - \langle \varphi _a , \, a \rangle | \le 1 + |a|\) hence
which (together with (77)) yields the upper estimate in (75). The lower estimate is analogous. Similar calculations show that
ending the proof of Theorem 1.
6 Topology of Alexander’s map
Let \(M = \{ (z,w) \in S^3 : w + \overline{w} > 0 \}\) (an open subset of \(S^3\)) and \(\Sigma \) a codimension 2 totally geodesic submanifold of \(S^5\). There is a choice of complex coordinates \((Z_1 , Z_2 , Z_3 )\) on \({\mathbb {C}}^3\) such that \(\Sigma = \{ Z \in S^5 : Z_3 = 0 \}\). A continuous map \(\phi : M \rightarrow S^5\) is said to meet \(\Sigma \) if \(\phi (M) \cap \Sigma \ne \emptyset \). If \(\phi \) doesn’t meet \(\Sigma \) then \(\phi \) is said to link \(\Sigma \) if the map \(\phi : M \rightarrow S^5 {\setminus } \Sigma \) is not homotopically null. The scope of this section is to prove
Theorem 3
Alexander’s map \(\phi (z,w) = (z^2 , \, \sqrt{2} \, z w, \, w^2 )\) links \(\Sigma \) as a map of \(M\) into \(S^5 {\setminus } \Sigma \).
Let \(\sigma = \{ (z,w) \in S^3 : w = 0 \}\) (so that \(\sigma = S^1 \times \{ 0 \}\)). The image of Alexander’s map \(\phi = \phi _{{\mathbb {A}}_0}\) intersects \(\Sigma \) along \(\phi (S^3 ) \cap \Sigma = \{ (Z \in {\mathbb {C}}^3 : Z_1 \in S^1 , \; (Z_2 , \, Z_3 ) = 0 \}\) so that \(M \subset S^3 {\setminus } \sigma \) (an open subset) and \(\phi : M \rightarrow S^5\) doesn’t meet \(\Sigma \). By a result of Solomon [27], for any harmonic map \(\phi \) from a compact Riemannian manifold into a sphere \(S^m\), \(\phi \) either meets or links a codimension two totally geodesic submanifold of \(S^m\). In view of Solomon’s result, it is a natural question whether \(\phi : M \rightarrow S^5 {\setminus } \Sigma \) is homotopically nontrivial. The beautiful proof in [27] (unfortunately confined to a contradiction loop) is to exploit \(S^{m-1}_+ \times _{\mathcal {W}} {\mathbb {R}} \rightarrow S^{m-1}_+ \times _{\mathcal {W}} S^1 \approx S^m {\setminus } S^{m-2}\) (a local isometry followed by an isometry) and harmonicity by using a particular 1-parameter variation of \(\phi \) which leads to a single linear elliptic equation on \(M\) (rather than a system of nonlinear elliptic equations) followed by applying Hopf’s maximum principle. The \({\mathbb {R}}\)-component of the lifted Alexander’s map \((F , u) : S^3 {\setminus } S^1 \rightarrow S^4_+ \times _{\mathcal {W}} {\mathbb {R}}\) is a solution to \(\Delta _b u = 0\) yet Solomon’s arguments do not carry over because \(S^3 {\setminus } S^1\) isn’t compact. As an attempt to circumnavigate the lack of compactness one may integrate \(\Delta _b (u^2 ) = 2 \, \Vert \nabla ^H u \Vert ^2\) over the domain \(\varOmega = \{ (z,w) \in S^3 {\setminus } S^1 : \mathrm{Im}(w) > 0 \}\) (mimicking the proof of Hopf’s maximum principle) and exploit \(u (\partial \varOmega ) \subset {\mathbb Z}\) to rid oneself of the boundary terms. Argument fails again because \(\partial \varOmega \) is not connected (it has two connected components \(S^2_\pm \times \{ 0 \}\)). To comply with this difficulty, we confine ourselves to the study of Alexander’s map \(\phi _{{\mathbb {A}}_0} : M = \{ (z,w) \in S^3 : \mathrm{Re}(w) > 0 \} \rightarrow S^5\) and integrate over the domain \(\varOmega = \{ (z,w) \in M : \mathrm{Im}(w) > 0 \}\). To prove Theorem 3 we shall need the \(C^\infty \) diffeomorphism
Here \(S^4_+ = \{ y \in S^4 : y_5 > 0 \}\). Let \(g_N\) denote the standard Sasakian metric on \(S^N \subset {\mathbb {R}}^{N+1}\) (so that \(g_3 = g_{\eta _0}\)). Let \({\mathcal {W}} : S^4_+ \rightarrow (0, + \infty )\) be the \(C^\infty \) function given by \({\mathcal {W}}(y) = y_5\). We endow \(S^4_+ \times S^1\) with the warped product metric \(h_{\mathcal {W}} = \pi _1^*g_4 + \left( {\mathcal {W}} \circ \pi _1 \right) ^2 \, \pi _2^*g_1\) (function \(\mathcal W\) is the warping factor). Here \(\pi _1 : S^4_+ \times S^1 \rightarrow S^4_+\) and \(\pi _2 : S^4_+ \times S^1 \rightarrow S^1\) are the projections. Pair \(\left( S^4_+ \times S^1 , \; h_{\mathcal {W}} \right) \) is commonly denoted by \(S^4_+ \times _{\mathcal {W}} S^1\) (and referred to as a warped product manifold). Diffeomorphism \(I\) is actually an isometry of \(S^4_+ \times _{\mathcal {W}} S^1\) onto \(\left( S^5 {\setminus } \Sigma , \, g_5 \right) \).
The remainder of the proof is by contradiction. Let us assume that \(\phi : M \rightarrow S^5 {\setminus } \Sigma \) is homotopically null, hence \(\psi = I^{-1} \circ \phi \) is homotopically null, as well. Thus \(\psi _*\, \pi _1 \left( M , x_0 \right) = 0\) where \(\pi _1 \left( M , x_0 \right) \) is the fundamental group of \(M\) with base point \(x_0 \in M\). If \(F = \pi _1 \circ \psi \) and \(f = \pi _2 \circ \psi \) (so that \(\psi = (F , f)\)) then \(f_*\, \pi _1 \left( M , x_0 \right) = 0\). Thus, given a point \(s_0 \in {\mathbb {R}}\) such that \(p(s_0 ) = f(x_0 )\), there is a unique continuous function \(u : M \rightarrow {\mathbb {R}}\) such that \(u(x_0 ) = s_0\) and \(p \circ u = f\). Here \(p : {\mathbb {R}} \rightarrow S^1\) is the exponential mapping \(p(s) = \exp (2 \pi i s)\). As
it follows that \(f(z,w) = (w /|w|)^2\) hence a posteriori \(u\) is smooth and (by differentiating in \(p \circ u = f\))
Let us endow \(S^4_+ \times {\mathbb {R}}\) with the warped product metric \(\hat{h}_{\mathcal {W}} = \varPi _1^*g_4 + \left( {\mathcal {W}} \circ \varPi _1 \right) ^2 \, \varPi _2^*d s^2\) where \(\varPi _1 : S^4_+ \times {\mathbb {R}} \rightarrow S^4_+\) and \(\varPi _2 : S^4_+ \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) are projections. Then \(1_{S^4_+} \times p\) is a local isometry of \(S^4_+ \times _{\mathcal {W}} {\mathbb {R}}\) onto \(S^4_+ \times _{\mathcal {W}} S^1\). Let \(\varOmega \subset M\) be given by \(\varOmega = \{ (z,w) \in M : (1/2i ) (w - \overline{w}) > 0 \}\) (a bounded domain in \(M\)). As \(\phi : M \rightarrow S^5 {\setminus } \Sigma \) is subelliptic harmonic and the maps \(I\) and \(1_{S^4_+} \times p\) are, respectively, an isometry and a local isometry, it follows that \((F, u) : M \rightarrow S^4_+ \times _{\mathcal {W}} {\mathbb {R}}\) is subelliptic harmonic, as well. Hence for every \(\varphi \in C^\infty _0 (\varOmega )\)
Note that
hence
for any \(|t| < \epsilon \). Consequently
so that
Next we observe that
and integrate by parts in (79) so that to obtain
for every \(\varphi \in C^\infty _0 (\varOmega )\). Therefore, \(u\) is a solution to the second-order subelliptic equation
in \(\varOmega \). By (78) \(\nabla ^H u = (1/\pi i) \left\{ ( z/\overline{w}) \, {{\mathcal {Z}}} - (\overline{z}/w) \, \overline{{\mathcal {Z}}} \right\} \) hence (exploiting \({\mathcal {W}} (F(z,w)) = |w|^2\))
so that [by (80)] \(\Delta _b u = 0\) in \(\varOmega \). Note that \(f(z,w) = 1\) for every \((z,w) \in \partial \varOmega \). Hence (by \(p \circ u = f\)) one has \(p(\partial \varOmega ) \subset {\mathbb Z}\). The boundary \(\partial \varOmega = S^2_+ \times \{ 0 \}\) is connected and \(u\) is continuous hence there is \(k \in {\mathbb Z}\) such that \(u(z,w) = k\) for every \((z,w) \in \partial \varOmega \). Let us recall the identity
Next we set \(\tilde{u} = u - k\) hence \(\Delta _b \tilde{u} = 0\) in \(\varOmega \). Integration over \(\varOmega \) in \(\Delta _b (\tilde{u}^2 ) = - 2 \Vert \nabla ^H \tilde{u} \Vert ^2\) furnishes
where \(C > 0\) is a constant such that \(\varPsi _{\eta _0} = C \, \mathrm{d} \, \mathrm{vol}(g_{\eta _0})\) and \(\iota : \partial \varOmega \rightarrow M\) is the inclusion. Thus \(\nabla ^H \tilde{u} = 0\) in \(\varOmega \), i.e., \(\tilde{u}\) is a real-valued CR function on \(\varOmega \). Consequently \(u\), and then \(f\), is constant on \(\varOmega \), a contradiction.
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Acknowledgments
The paper was partially written during Sorin Dragomir’s visit at the Department of Mathematics of University of Illinois at Urbana-Champaign (April 2013). He wishes to express his gratitude for the excellent hospitality and working conditions there and for inspiring discussions with Professor John D’Angelo. E. Barletta and S. Dragomir acknowledge support from Italian government through P.R.I.N. 2012 Varietà Reali e Complesse: Geometria, Topologia e Analisi Armonica.
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Barletta, E., Dragomir, S. Proper holomorphic maps in harmonic map theory. Annali di Matematica 194, 1469–1498 (2015). https://doi.org/10.1007/s10231-014-0429-z
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DOI: https://doi.org/10.1007/s10231-014-0429-z