Research Article | Open Access
On the Regularity of Weak Contact -Harmonic Maps
We prove Caccioppoli type estimates and consequently establish local Hölder continuity for a class of weak contact -harmonic maps from the Heisenberg group into the sphere .
The study of pseudoharmonic maps was started by Barletta et al.  (cf. also [2, 3] for successive investigations) as a generalization of the theory of harmonic maps among Riemannian manifolds (cf., e.g., ) and by identifying the results of Jost and Xu , Zhou , Hajłasz and Strzelecki , and Wang  as local aspects of the theory of pseudoharmonic maps from a strictly pseudoconvex CR manifold into a Riemannian manifold (cf. also [9, pages 225-226]).
A similar class of maps, yet with values in another CR manifold, was studied in . These are critical points of the functional where is a compact strictly pseudoconvex CR manifold of CR dimension , , and is a contact form on . Also is a contact Riemannian manifold and in particular an almost CR manifold (of CR codimension ).
A moment's thought reveals the augmented difficulties such a theory may present. For instance, if and are two strictly pseudoconvex CR manifolds endowed, respectively, with contact forms and , then the pseudohermitian analog of the notion of a harmonic morphism (among Riemannian manifolds) is quite obvious: one may consider continuous maps such that the pullback of any local solution to in satisfies in in distribution sense. Here and are the sublaplacians of and , respectively. Unlike the situation in  (where the target manifold is Riemannian and pulls back local harmonic functions on to distribution solutions of ) such is not necessarily smooth (since it is unknown whether local coordinate systems on such that in might be produced). To give another example, should one look for a pseudohermitian analog to the Fluglede-Ishihara theorem (cf.  when is CR and is Riemannian), one would face the lack of an Ishihara type lemma (cf. ) as it is unknown whether admits local solutions whose (horizontal) gradient and hessian have prescribed values at a point. Moreover, what would be the appropriate notion of a hessian (cf.  for a possible choice)?
A third example, discussed at some length in this paper, is that of the “degeneracy” of the Euler-Lagrange equations associated to the variational principle when is a Sasakian manifold. Indeed the matrix has but rank at each point (a well-known phenomenon in contact Riemannian geometry, cf., e.g., . See also ). Consequently, in general one may not expect regularity of weak solutions to (2). For instance, if is the Heisenberg group and is a solution to (2), then is subject to yet is an arbitrary function (cf. Section 3). For the more appealing case, where is the Heisenberg group and is the sphere, (2) may be written as (cf. Proposition 15) which is indeed the form assumed by the Euler-Lagrange equations in , yet unlike the situation there in general (cf. Proposition 16 for the notations). Although has a quite explicit form (yielding—for a class of weak solutions which are close to being horizontal maps—simple estimates on ), only a weaker form of the duality inequality lemma in  may be proved (cf. Lemma 17) leading nevertheless (together with a hole filling argument) to Caccioppoli type estimates for some and , which are known (cf., e.g.,  for a very general argument based on work in ) to imply the local Hölder continuity of the given weak solution.
The paper is organized as follows. In Section 2 we recall a few conventions and basic results obtained in . Sections 3 and 4 are devoted to the study of the local properties of weak contact -harmonic maps. We show that weak contact -maps are locally Hölder continuous (cf. Corollary 21) provided they are close to being horizontal maps; that is, the assumptions (96) are satisfied. The relevance of the number stems from the facts that is a CR invariant and is the homogeneous dimension of . The authors believe that subelliptic theory should play within CR geometry, as a branch of complex analysis in several complex variables, the strong role played by elliptic theory in Riemannian geometry, and the present paper is a step in this direction.
2. Basic Conventions and Results
For all notions of CR and pseudohermitian geometry we adopt the conventions and notations in the monograph . For the approach to contact structures within Riemannian geometry we rely on the presentation in Blair , (cf. also Tanno ). Given a real -dimensional differentiable manifold , an almost CR structure is a complex subbundle of the complexified tangent bundle, of complex rank , such that for any . Here and overbars indicate complex conjugates. The integer is the CR dimension of the almost CR manifold . Almost CR structures are a bundle theoretic recast of the tangential Cauchy-Riemann operator given by for any and any . An almost CR structure is (formally or Frobenius) integrable if for any and any open set . The tangential C-R operator may be extended to arbitrary -forms on and the resulting pseudocomplex , , is a complex (i.e., ) if and only if the given almost CR structure is integrable (cf. ). Integrable almost CR structures are commonly referred to as CR structures and appear mainly on real hypersurfaces of complex manifolds, as induced by the complex structure of the ambient space; that is, for any complex manifold and any real hypersurface is a CR structure on . Here is the holomorphic tangent bundle over (locally the span of for any local system of complex coordinates on ). Also is the complex dimension of , and then the CR dimension of is . Integrability of (7) follows from the Nijenhuis integrability of the complex structure on . A solution to (the tangential C-R equations) is a CR function on and, in the context of real hypersurfaces carrying the induced CR structure (7), CR functions appear as traces on of holomorphic functions defined on a neighborhood of in . Hence to say that the CR structure is given by (7) is to say that the tangential C-R equations are induced by the ordinary Cauchy-Riemann system on . CR functions which are not traces of holomorphic functions may exist (cf., e.g., ). CR structures which are not given by (7), and for which there is not any embedding of into some complex manifold yielding (7), do exist as well (cf. again [17, page 172]). An array of geometric objects, such as pseudohermitian structures, the Levi form (cf. [9, 18]) and successively (in the nondegenerate case) contact structures, the Tanaka-Webster connection (cf. [18, 19]), the sublaplacian and the Fefferman metric (cf. [9, 20]), springs from the given CR structure very much the way the complex structure determines the metric structure (up to a conformal invariant) on a Riemann surface and are thought of as geometric tools whose use will ultimately shed light on the properties of solutions, local and global, to the tangential C-R equations. Integrability of appears as a built-in ingredient of objects such as the Tanaka-Webster connection or the Fefferman metric, yet it is believed to lack the geometric meaning of involutivity of real smooth distributions on manifolds (cf., e.g., [21, page 16]). On the other hand nonintegrable examples of almost CR structures occur frequently, either on real hypersurfaces of almost complex manifolds or on contact Riemannian manifolds (cf. [13, 16]). A remedy was indicated by Tanno , showing that the wealth of additional structure on a given contact Riemannian manifold compensates for the lack of integrability of and specifically providing a generalization of the Tanaka-Webster connection to the nonintegrable context.
Given a CR manifold , let be the Levi, or maximally complex, distribution and , , its complex structure. Let , , be the conormal bundle associated to , a real line bundle over . Since is assumed to be connected and orientable, the conormal bundle is trivial. A globally defined nowhere zero section is a pseudohermitian structure on . For each pseudohermitian structure on the Levi form is Two pseudohermitian structures are related by for some function . If this is the case, then . A CR manifold is nondegenerate (resp., strictly pseudoconvex) if is nondegenerate (resp., positive definite) for some . If is a nondegenerate CR manifold, of CR dimension , then each pseudohermitian structure is a contact form; that is, is a volume form on . If is nondegenerate and is a contact form on , there is a unique globally defined, nowhere zero, tangent vector field (the Reeb vector field of ()) such that and . The Webster metric is the semi-Riemannian metric on given by for any . If is strictly pseudoconvex and is chosen such that is positive definite, then is a Riemannian metric on .
Let be a -dimensional manifold (). An almost contact structure on is a synthetic object consisting of a -tensor field , a vector field , and a -form such that with respect to any local coordinate system on . A Riemannian metric on is associated, or compatible, to the almost contact structure (and is an almost contact metric structure on ) if Associated metrics always exist (cf. ). A contact metric structure is an almost contact metric structure such that , where is the -form given by .
Let be a map from a strictly pseudoconvex CR manifold of CR dimension into a contact Riemannian manifold . Let be a contact form on such that the Levi form is positive definite. Let and let us consider the vector bundle valued form given by where is the natural projection associated to the decomposition . Let and let be a local -orthonormal frame of defined on an open neighborhood of . We set Note that
Definition 1. Let . A map is said to be contact -harmonic if is a critical point of the energy functional for any relatively compact domain . Contact -harmonic maps are called contact harmonic maps.
Let be the Tanaka-Webster connection of that is the unique linear connection on obeying to (i) is -parallel (i.e., for any and any ), (ii) and , and (iii) the torsion tensor field of is pure (i.e., , for any and , where for any (cf. Theorem 1.3 and Definition 1.25 in [9, pages 25-26]). The vector valued -form is the pseudohermitian torsion of . Let be the generalized Tanaka-Webster connection of given locally by (cf., e.g., ), where are the Christoffel symbols of . Covariant derivatives are meant with respect to the Levi-Civita connection of . For each we consider given by Let be the connection induced by in the pullback bundle . We set Let and let be a local -orthonormal frame of defined on an open neighborhood of . We define a section in by setting where denotes the restriction of to . By a result in  the Euler-Lagrange equations associated to the variational principle are here (cf., e.g., ). Also is the pseudohermitian torsion of ; that is, , and for any . are again the local coefficients of with respect to . In particular if is a Sasakian metric, then is contact -harmonic if and only if
3. Weak Contact Harmonic Maps
Sections 3 and 4 are devoted to the study of local properties of weak critical points of the functional (15). A study of the regularity of weak solutions to subelliptic systems (such as (53)) was started by Wang , and Capogna and Garofalo , though only for maps from Carnot groups, (cf. also Zhou ).
Let be a strictly pseudoconvex CR manifold and a contact form on . Let be a local -orthonormal frame of defined on the open set and the formal adjoint of ; that is, where and . Also are the local coefficients of the Tanaka-Webster connection of with respect to the local coordinate system on . Clearly for any , where .
Proposition 2. Let be a smooth map and a Sasakian metric on . Then is contact -harmonic if and only if for any local orthonormal frame of .
Example 3 (contact -harmonic maps into the Heisenberg group). Let , , be the Heisenberg group (cf., e.g., [9, pages 11–14]). Let be the Cartesian coordinates on and let Let be the -tensor field on determined by where . Next the differential -form given by is a contact form on ; that is, is a volume form. Let . Finally we shall need the Riemannian metric on given by on , on , and . Then is a Sasakian metric on (and actually is a Sasakian space form of -sectional ; cf., e.g., ). A calculation shows that where and . Let and let be the span of over . Then is a strictly pseudoconvex CR structure on and is a contact form such that the Levi form is positive definite. Let be the Tanaka-Webster connection of . A calculation shows that where and for simplicity. Hence and the remaining connection coefficients vanish. The Webster metric of is given by hence (by a straightforward calculation) where and . Let us substitute (28)–(32) into (23) so that to obtain Hence if is a contact -harmonic map, then is subject to (33) while is an arbitrary function. Therefore, in general one may not expect regularity for a given (weak) contact -harmonic map.
Lemma 4. A smooth map of a strictly pseudoconvex CR manifold into a Sasakian manifold is contact -harmonic if and only if for any local orthonormal frame of on and any local coordinate system on such that .
Proof. Let us multiply (23) by a test function and integrate by parts On the other hand (as both and are parallel with respect to ) where are the coefficients of with respect to . Therefore (35) may be written as and Lemma 4 is proved.
Let us consider the function spaces where are understood as weak derivatives. If , then are separable Banach spaces with the norms Also is reflexive provided that . The central concept of this section may be introduced as follows. Let be a -orthonormal frame of defined on the open set . Let be an open set which is relatively compact in a larger coordinate neighborhood in .
Example 6 (Example 3 continued). A weak solution to (33) is a map such that and for any . We need to recall the following general result, due to Xu and Zuily . Let be a Hörmander system on an open set , , and a domain such that . Let be a symmetric and positive definite matrix defined in . If for any , then any continuous solution to in is actually smooth. Let us assume that is a domain such that is contained in a coordinate neighborhood in . By the result in  quoted above.
Proposition 7. For any weak solution to the contact -harmonic map system (33) if , then .
Of course in the particular case any distribution solution is (as the operator is hypoelliptic).
Example 8 (contact -harmonic maps into the sphere). Let and let be the canonical Sasakian metric on . Then a contact -harmonic map is a solution to for any . Here and with , . Equation (46) follows from (23) by computing the Christoffel symbols of with respect to the local coordinate system that is so that On the other hand (cf. ) so that for any Sasakian metric . When , the identities (49)–(51) lead to and then to (46) by taking into account that is an -structure on ; that is, . Our next purpose in this example is to prove the following result.
Proposition 9. Let be a horizontal map. Then is contact -harmonic if and only if is subelliptic -harmonic with respect to the canonical Hörmander system on .
According to  given a Hörmander system of vector fields defined on an open set , one may adopt the following.
Definition 10. A subelliptic -harmonic map is a solution to the system (the formal adjoint of in  is under the conventions adopted in the present paper) such that .
A horizontal map is a smooth map such that One may define weak solutions to (54) by requiring that for some and that (54) holds a.e. in . Then the statement in Proposition 9 holds for weak solutions of the relevant equations as well. In particular, by a result in , any weak horizontal contact -harmonic map is locally Hölder continuous provided that .
Lemma 11. Let be a strictly pseudoconvex CR manifold. A smooth map is contact -harmonic if and only if for any and any local orthonormal frame of .
By (14) if is a horizontal map, then and one may readily check that (55) is equivalent to (53) for any . Of course the component will satisfy (53) as well (as a consequence of the constraint ). To prove Lemma 11, let us multiply (46) by a test function and integrate over . The left-hand side of the resulting equation is where . Then (by (37)) hence (46) implies which yields (55) because on the sphere Lemma 11 is proved.
The notion of a weak contact harmonic map as introduced above is confined to maps such that the target contact Riemannian manifold is covered by a single coordinate neighborhood. Another natural approach (customary in the theory of harmonic maps among Riemannian manifolds, cf., e.g., [4, page 38]) is to use Nash's embedding theorem (cf. ) in order to embed isometrically the target manifold into some Euclidean space and produce an alternative first variation formula (cf. Theorem 2.22 in [26, page 139]) depending however on the embedding .
A generalization of Nash's embedding theorem to the context of contact Riemannian geometry has been obtained by D'Ambra . Let be the Heisenberg group equipped with the standard Sasakian structure . Let be a contact Riemannian manifold. By a result in , if is compact and , there is a -embedding which is both horizontal, that is, , and isometric in the sense that preserves the Levi forms Any contact Riemannian manifold is in particular a sub-Riemannian manifold (in the sense of ); hence carries the Carnot-Carathéodory metric associated to the sub-Riemannian structure . In particular is an isometry among the metric spaces and (cf. Section 7 for the definition of the distance function ). As also possesses a linear space structure, the methods in  (methods of direct infinitesimal geometry) become available on a contact Riemannian manifold (e.g., one may merely use the balls with respect to and the linear structure of the ambient space to reformulate on Definition 2.1 in [29, page 280]) and we conjecture that the arguments in  may be recovered to study the equation on a strictly pseudoconvex CR manifold (the theory in  only deals with second order degenerate elliptic equations on domains in ). Unfortunately the existence of -embeddings of given contact structures is not sufficient for differential geometric purposes, as long as Gauss and Weingarten formulae (which require two derivatives of ) are involved. The problem of improving D'Ambra's proof (to get a horizontal embedding of class at least ) is open.
4. Contact Harmonic Maps into Spheres
Let be a bounded open set and a Hörmander system of vector fields such that . We recall (cf., e.g., [9, page 261]) the following.
Definition 12. A number is a homogeneous dimension relative to with respect to if there is a constant such that for any Carnot-Carathéodory ball of center and radius and any Carnot-Carathéodory ball of center and radius .
The diameter of is meant with respect to the Carnot-Carathéodory metric associated to . Hajłasz and Strzelecki  studied local properties of weak solutions to the system (53). Their main finding is that every weak subelliptic -harmonic map (i.e., every weak solution to (53) with ) is locally Hölder continuous. Maps with values in a unit sphere have a special status due to the fact that the subelliptic harmonic map system (here (53)) may be written in a simple form using an approach commonly referred to as the Frédéric Hélein trick (cf. [7, page 353], see also Hélein ). The purpose of this section is to start a study of weak solutions to the system (55) following the ideas in  though confined to maps which are “close to horizontal” in a sense to be made precise in the sequel.
Let be the Heisenberg group equipped with the standard contact form . Let be a bounded domain. Let be the -orthonormal frame given by and , where as in Example 3. Clearly the coefficients of the 's lie in . We recall that an absolutely continuous curve is admissible if for some functions such that .
Definition 13. The Carnot-Carathéodory distance among two points is the infimum of all for which there exists an admissible curve such that and . Balls with respect to are denoted by and referred to as Carnot-Carathéodory balls.
We shall characterize horizontal maps in terms of the first order differential operator defined for .
Proposition 14. Let be a map such that for any . Then is a (weak) horizontal map if and only if for any .
Let be the natural complex coordinates on and set and . The following conventions are adopted as to the range of indices: Let so that the pointwise restriction of to is a unit normal field on . Let be the complex structure on . Then given by for any is the Reeb vector field on . Here is the inclusion. With respect to the local chart in Example 8 the Reeb vector is given by Then together with (48) in Example 8 leads to Finally (66) implies that . Proposition 14 is proved. In particular may be written as Our next task is to put (55) into a more tractable form.
Proposition 15. Let such that . Let us consider the functions with . Let for any . Then is a contact -harmonic map if and only if
Here the dot product means . Using and (65) and (66), one obtains Then substitution into (55) leads to It remains to be shown that (71) and (72) imply Let us multiply (71) by , where is an arbitrary test function, and integrate over so that to obtain (after integration by parts) Similarly let us multiply (72) by so that to obtain Let us contract the indices and in (74) (resp., and in (75)), add the resulting equations, and use the identities