On the real-analyticity of rigid spherical hypersurfaces in C 2
aa r X i v : . [ m a t h . C V ] M a r ON THE REAL-ANALYTICITY OF RIGIDSPHERICAL HYPERSURFACES IN C ALEXANDER ISAEV AND JO¨EL MERKER
Abstract.
We prove that every smooth rigid spherical hypersurface in C isin fact real-analytic. As an application of this result, it follows that the classi-fication of real-analytic rigid spherical hypersurfaces in C found by V. Ezhovand G. Schmalz applies in the smooth case. Introduction
We consider connected smooth real hypersurfaces in the space C with standardcoordinates z “ x ` iy, w “ u ` iv . For simplicity, we understand smoothness as C -smoothness although everything we do works for the class C k with sufficientlylarge k , say for the class C . Specifically, we look at rigid hypersurfaces , i.e.,hypersurfaces given by equations of the form(1.1) u “ F p z, z q , where F is a smooth real-valued function defined on a domain U Ă C . Rigid hyper-surfaces are invariant under the 1-parameter family of holomorphic transformations(1.2) p z, w q ÞÑ p z, w ` it q , t P R . Throughout the paper we assume Levi-nondegeneracy, which for a rigid hyper-surface means that F z ¯ z is everywhere nonvanishing (here and below z, ¯ z used assubscripts indicate the corresponding partial derivatives).All our considerations will be entirely local, in a neighborhood of a chosen point.In particular, letting 0 be the reference point, we utilize a natural notion of localequivalence for germs of rigid hypersurfaces. Namely, two rigid hypersurface germsat 0 are called rigidly equivalent if there exists a map of the form p z, w q ÞÑ p g p z q , aw ` h p z qq , a P R ˚ that transforms one germ into the other, where g, h are holomorphic near 0 and g p q ‰ g p q “ h p q “ p˚q classify, up to rigid equivalence, the germs of rigid hypersurfaces that are spherical , i.e., CR-equivalent to a germ of the sphere S Ă C .The sphericity condition for a Levi-nondegenerate smooth hypersurface M Ă C is equivalent to that of the vanishing of the CR-curvature, which is a su p , q -valued2-form defined on an 8-dimensional principal fiber bundle over M . This form arisesfrom the reduction of the CR-structures of 3-dimensional Levi-nondegenerate CR-manifolds of hypersurface type to absolute parallelisms performed in [Ca] and latergeneralized to higher dimensions, in particular, in papers [T1], [T2], [T3], [CM], Mathematics Subject Classification:
Keywords: rigid hypersurfaces, zero CR-curvature equation, nonlinear elliptic systems, real-analyticity of solutions.
ISAEV AND MERKER [Ch], [BS2], [BS3]. The interpretation of the sphericity condition as that of CR-flatness is a basis of our approach to task p˚q .The problem of describing CR-flat structures that possess certain symmetries(e.g., as in (1.2)) is a natural one and has been addressed by a number of authors. Inparticular, under the assumption of CR-flatness, homogeneous strongly pseudocon-vex CR-hypersurfaces have been studied (see [BS1]), and tube
Levi-nondegeneratehypersurfaces in complex space have been extensively investigated and even fullyclassified for certain signatures of the Levi form (see [I2] for a detailed exposition).Furthermore, in [I3] a class of CR-flat
Levi-degenerate tube hypersurfaces in C wasfully described. Compared to the tube case, the case of rigid hypersurfaces is thenext situation up in terms of complexity and is substantially harder. For instance,as explained in [I4], the study of Levi-degenerate rigid hypersurfaces in C is a muchmore difficult task in comparison with the work on tube hypersurfaces done in [I3].Although not mentioned explicitly, problem p˚q was first looked at in article [S]for real-analytic hypersurfaces. Even in this simplified setup, determining all rigidspherical hypersurfaces turned out to be highly nontrivial, with only a number ofexamples found in [S]. A complete solution to p˚q in the real-analytic category wasonly recently obtained in [ES] by employing techniques based on the rigid normalforms constructed in [S]. Although the hypersurface germs on the list found in[ES] are given by implicit equations, these equations are not very complicated andrepresent an acceptable solution to problem p˚q in the real-analytic situation. Forthe completeness of our exposition, we state the result of [ES] in Section 3 (seeTheorem 3.1).In this note we show that the classification of [ES] applies in the smooth case aswell. Namely, we obtain the following: THEOREM 1.1.
Every smooth rigid spherical hypersurface in C is real-analytic. Our proof of Theorem 1.1 in the next section is based on the zero CR-curvatureequations for rigid Levi-nondegenerate hypersurfaces in C n for any n ě F in C (see formula (2.1) in Theorem 2.1 below). We note that another PDE for thefunction F , also characterizing the sphericity property of a rigid Levi-nondegeneratehypersurface in C , was produced in [L] (see formula (2.4)). The argument of [L]is based on the Chern-Moser normal forms, so it assumes real-analyticity. In fact,equation (2.4) can be shown to characterize sphericity even in the smooth setup,hence it is equivalent to (2.1). One way to see this is by specializing calculationsin articles [MS1], [MS2] to the rigid case. Thus, each of (2.1), (2.4) can be used toestablish Theorem 1.1, but the argument based on equation (2.1) is shorter, so wechose to utilize (2.1) in our proof. We sketch a proof relying on equation (2.4) inRemark 2.2. Acknowledgements.
This work was initiated while the first author was visitingD´epartement de Math´ematiques d’Orsay and completed while he was visiting theSteklov Mathematical Institute in Moscow. We thank S. Nemirovski and A. Domrinfor useful discussions. 2.
Proof of Theorem 1.1
We start by stating the special case of [I2, Theorem 2.1] in complex dimensiontwo.
THEOREM 2.1.
Consider a rigid hypersurface M in C given by equation (1.1) with the function F defined on a domain U Ă C . Assume that F is everywhere EAL-ANALYTICITY OF RIGID SPHERICAL HYPERSURFACES 3
Levi-nondegenerate, i.e., F z ¯ z ‰ on U . Then M is spherical if and only if F satisfies a differential equation of the form (2.1) F zz “ A p F z q ` B p F z q ` CF z ` D, where A , B , C , D are functions holomorphic on U . Set f : “ F z . To prove Theorem 1.1 it suffices to show that f is real-analytic on U . Indeed, if f were real-analytic, then locally near every z P U we would have F “ ż f dz ` ϕ p ¯ z q , where ş f dz denotes the result of the term-by-term integration with respect to z ´ z of the power series in z ´ z , z ´ z representing f near z , and ϕ is antiholomorphichence real-analytic. It would then follow that F is real-analytic at z and thereforeon U .To prove that f is real-analytic on U , observe that (2.1) implies(2.2) f z “ Af ` Bf ` Cf ` D. Separating the real and imaginary parts of equation (2.2), we see that it is equivalentto a system of two equations for two real-valued functions:(2.3) r x ` s y “ G p z, ¯ z, r, s q , ´ r y ` s x “ H p z, ¯ z, r, s q , where r : “ Re f , s : “ Im f and G , H are real-valued analytic functions of all theirarguments. Clearly, system (2.3) is nonlinear in general. Sincedet ˆ λ λ ´ λ λ ˙ “ λ ` λ is positive for all nonzero vectors p λ , λ q P R , system (2.3) is elliptic as definedin [M2, pp. 210, 266]. Then by [M1, p. 203] (see also [M2, Theorem 6.7.6]), everysolution of (2.3) is real-analytic. We thus see that f is real-analytic as required. l Remark . As mentioned in the introduction, in article [L] another equationcharacterizing the sphericity property of a real-analytic rigid Levi-nondegeneratehypersurface in C was obtained, and, by results of [MS1], [MS2], this equationworks in the smooth setup as well. Let µ : “ log p F z ¯ z q ¯ z “ F z ¯ z ¯ z F z ¯ z . Then the equation found in [L] is as follows:(2.4) µ z ¯ z ¯ z ´ µ z ¯ z µ ` µ z µ ´ µ z µ ¯ z “ . We will now show that (2.4) leads to an elliptic system, just as (2.2) does.Indeed, separating the real and imaginary parts of equation (2.4), we see that it isequivalent to a system of two equations for two real-valued functions:(2.5) ν xxx ` ν xyy ´ η xxy ´ η yyy “ Φ ,ν xxy ` ν yyy ` η xxx ´ η xyy “ Ψ , where ν : “ Re µ , η : “ Im µ and Φ, Ψ are analytic functions of ν , η , as well as theirfirst and second partial derivatives with respect to x , y . To see that system (2.5)is elliptic, we computedet ˜ λ ` λ λ ´ λ λ ´ λ λ λ ` λ λ ´ λ λ ¸ “ λ ` λ λ ` λ λ ` λ , ISAEV AND MERKER which is positive for all nonzero vectors p λ , λ q P R . Hence, as before, by [M1,p. 203] and [M2, Theorem 6.7.6] every solution of (2.5) is real-analytic, and therefore µ is real-analytic. It then follows that locally near every z P U we have F z ¯ z “ exp ˆż µd ¯ z ˙ exp p ϕ p z qq , where ϕ is holomorphic. Thus, F z ¯ z is real-analytic at z , and, by the ellipticity ofPoisson’s equation, F is real-analytic on U .3. The Ezhov-Schmalz classification
In this section, for the completeness of our exposition, we state the main resultof [ES], which is a solution to problem p˚q in the real-analytic setup.
THEOREM 3.1.
Every germ at the origin of a real-analytic rigid spherical hyper-surface in C is rigidly equivalent to the germ at the origin of the rigid hypersurfacedefined by an equation of the form (3.1) p ` φ | z | q sin p ru q r ´ e ´ θu | z | ´ ` φ ` ¯ cz ` c ¯ z ` φ p φ ´ θ q| z | ˘ e ´ θu ´ cos p ru q ` θ sin p ru q r r ` θ “ . Here φ, θ P R , c P C , r P R Y i R and there exists a pair of numbers τ, ρ P R suchthat: φ is any real root of the cubic polynomial (3.2) 4 φ ` τ φ ` p τ ´ ρ q φ ´ | c | , and θ “ τ ` φ,r “ φ ` τ φ ´ ρ. In formula (3.1) , by expanding into power series and taking limits, one sets sin p ru q r ˇˇˇˇˇ r “ : “ u and e ´ θu ´ cos p ru q ` θ sin p ru q r r ` θ ˇˇˇˇˇ r “ θ “ : “ u . Notice that every equation in (3.1), up to multiplying r by -1 and up to thethree possible choices of φ as a real root of (3.2), is uniquely determined by threenumbers: c P C , τ P R , ρ P R . Multiplication of r by -1 leads to a rigidly equivalenthypersurface germ, whereas it is not clear whether different choices of φ can resultin some nonequivalent germs. However, even if polynomial (3.2) has a unique realroot, two distinct triplets p c, τ, ρ q and p c , τ , ρ q do not necessarily yield rigidlynonequivalent hypersurface germs. For example, for c “ τ “ ρ ă φ “ θ “ r “ ˘?´ ρ . In this case formula (3.1) becomessin p˘ ?´ ρ u q˘ ?´ ρ ´ | z | “ . By scaling z, w one observes that the germ at 0 of this hypersurface for any ρ ă u “ | z | , EAL-ANALYTICITY OF RIGID SPHERICAL HYPERSURFACES 5 which is one of the examples found in [S]. Thus, Theorem 3.1 does not describe themoduli space of real-analytic rigid spherical hypersurface germs in C precisely.We conclude our paper with the following consequence of Theorem 1.1: Corollary 3.2.
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