CONVEXITY OF THE ENTROPY OF POSITIVE SOLUTIONS TO THE HEAT EQUATION ON QUATERNIONIC CONTACT AND CR MANIFOLDS

Transcription

1 CONVEXITY OF THE ENTROPY OF POSITIVE SOLUTIONS TO THE HEAT EQUATION ON QUATERNIONIC CONTACT AND CR ANIFOLDS D. VASSILEV Abstract. A proof of the monotonicity of an entropy like energy for the heat equation on a quaternionic contact and CR manifolds is given. Contents. Introduction 2. Proofs of the Propositions Some preliminaries Proof of Proposition Proof of Proposition The CR case 8 References 0. Introduction The purpose of this note is to show the monotonicity of the entropy type energy associated to the (sub-elliptic) heat equation in a sub-riemannian setting. The result is inspired by the corresponding Riemannian fact related to Perelman s entropy formula for the heat equation on a static Riemannian manifold, see 8]. ore recently a similar quantity was considered in the CR case 4]. Our goal is to prove the convexity of the entropy of a positive solution to the (sub-elliptic) heat equation on a quaternionic contact manifold, henceforth abbreviated to qc, and give an alternative proof of the result in 4], more in line with the Riemannian case. We resolve directly the difficulties arising in the sub-riemannian setting of both quaternionic contact and CR manifolds. Section 3 contains the alternative proof of the result of 4] in the CR case while the remaining parts of the paper focus on the qc case. To state the problem, let be a quaternionic contact or a CR manifold of real dimensions +3 and 2n +, respectively, and u be a smooth positive solution to the (qc or CR) heat equation (.) t u = u. Hereafter, u = tr g ( 2 u) denotes the negative sub-laplacian with the trace taken with respect to an orthonormal basis of the respective horizontal or 2n dimensional spaces in the qc and CR Date: ay 0, athematics Subject Classification. 53C2,58J60,53C7,35P5,53C25. Key words and phrases. quaternionic contact structures, entropy monotonicity, Paneitz operator, 3-Sasakian.

2 2 D. VASSILEV cases. Associated to such a solution are the (Boltzmann-Nash like) entropy (.2) N(t) = u ln u V ol η and entropy energy functional (.3) E(t) = f 2 u V ol η, where, as usual, f = ln u and V ol η is the naturally associated volume form on, see (2.4) and (3.3). Exactly as in the Riemannian case, we have that the entropy is decreasing (i.e., nonincreasing) because of the formula d dt N = E(t). Our goal is the computation of the second derivative of the entropy. In order to state the result in the qc case we consider the Ricci type tensor (.4) L(X, X) def = 2Sg(X, X) + α n T 0 (X, X) + β n U(X, X), where X is any vector from the horizontal distribution, α n = 2(2n+3) 2n+, β n = 4(2n )(n+2) (2n+)(n ), and T 0 and U are certain invariant components of the torsion, see Subsection 2.. The tensor (.4) appeared earlier as a natural assumption in the qc Lichnerowicz-Obata type results, see 7, Section 8.] and references therein. In addition, following 2], we define the P form of a fixed smooth function f on by the following equation (.5) P f (X) = b= 3 f(x, e b, e b ) + t= b= 3 f(i t X, e b, I t e b ) Sdf(X) + T 0 (X, f) 8n(n 2) U(X, f), n which in the case n = is defined by formally dropping the last term. The P function of f is the function P f ( f). The C operator of is the 4-th order differential operator f Cf = P f = a= ( ea P f ) (e a ). In many respects the C operator plays a role similar to the Paneitz operator in CR geometry. We say that the P function of f is non-negative if f Cf V ol η = P f ( f) V ol η 0. If the above holds for any f C o () we say that the C operator is non-negative, C 0. We are ready to state our first result. Proposition.. Let be a compact QC manifold of dimension + 3. If u = e f is a positive solution to heat equation (.), then we have 2n + E (t) = ( 2 f) n + L( f, f) + ] 2 6n f 4 u V ol η + 3 P F ( F ) V ol η, n where u = F 2 (f = 2 ln F ) and ( 2 f) 0 is the traceless part of horizontal Hessian of f.

3 ENTROPY CONVEXITY ON QC AND CR ANIFOLDS 3 Several important properties of the C-operator were found in 2], most notable of which is the fact that the C operator is non-negative for n >. In dimension seven, n =, the condition of nonnegativity of the C operator is non-trivial. However, 2] showed that on a 7-dimensional compact qc-einstein manifold with positive qc-scalar curvature the P function of an eigenfunction of the sub-laplacian is non-negative. In particular, this property holds on any 3-Sasakian manifold, see 0, Corollary 4.3]. Clearly, these facts together with Proposition. imply the following theorem. Theorem.2. Let be a compact QC manifold of dimension + 3 of non-negative Ricci type tensor L(X, X) 0. In the case n = assume, in addition, that the C operator is non-negative. If u = e f is a positive solution of the heat equation (.), then the energy E(t) is monotone decreasing (i.e., non-increasing). The proof of Proposition. follows one of L. Ni s arguments 8] in the Riemannian case, thus it relies on Bochner s formula. ore precisely, after Ni s initial step, in order to handle the extra terms in Bochner s formula, we will follow the presentation of 7] where this was done for the qc Lichnerowicz type lower eigenvalue bound under positive Ricci type tensor, see 3, 2] for the original result. In the qc case, similar to the CR case, the Bochner formula has additional hard to control terms, which include the P -function of f. In our case, since the integrals are with respect to the measure u V ol η, rather than V ol η as in the Lichnerowicz type estimate, some new estimates are needed. The key is the following proposition which can be considered as an estimates from above of the integral of the P -function of f with respect to the measure u V ol η when the C operator is non-negative. Proposition.3. Let (, η) be a compact QC manifold of dimension + 3. If u = e f is a positive solution to heat equation (.), then with f = 2 ln F we have the identity (.6) P f ( f)u V ol η = f 4 u V ol η + 4 P F ( F ) V ol η. 4 In the last section of the paper we apply the same method in the case of a strictly pseudoconvex pseudohermitian manifold and prove the following Proposition. Proposition.4. Let be a compact strictly pseudoconvex pseudohermitian CR manifold of dimension 2n +. If u = e f is a positive solution to the heat equation (.), then we have n + 2n E (t) = ( 2 f) n + L( f, f) + ] 2 8n f 4 u V ol η 6 F C(F ) V ol η, n where u = F 2, ( 2 f) 0 is the traceless part of horizontal Hessian of f and C is the CR-Paneitz operator of. We refer to Section 3 for the relevant notation and definitions. As a consequence of Proposition.4 we recover the monotonicity of the entropy energy shown previously in 4]. We note that one of the motivations to consider the problem was the application of the CR version of the monotonicity of the entropy like energy 4, Lemma 3.3] in obtaining (non-optimal) estimate on the bottom of the L 2 spectrum of the CR sub-laplacian. However, the proof of 4, Corollary.9 and Section 6] is not fully justified since 4, Lemma3.3] is proved for a compact manifold. It should be noted that a proof of S-Y Cheng s type (even non-optimal) estimate in a sub-riemannian setting, such as CR or qc-manifold, is an interesting problem in particular because of the lack of general comparison theorems. We conclude by mentioning another proof of the monotonicity of the energy in the recent preprint ], which was the result of a past collaborative work with Ivanov and Petkov. Remarkably, 4]

4 4 D. VASSILEV is also not acknowledged in ] despite the line for line substantial overlap of, Section 3] with Chang and Wu proof 4, Lemma 3.3]. In this paper we give an independent direct approach to the problem. Acknowledgments. Thanks are due to Qi Zhang for insightful comments on the S-Y Cheng eigenvalue estimate during the Beijing Workshop on Conformal Geometry and Geometric PDE in 205, and to Jack Lee and Ben Chow for some useful discussions. The author also acknowledges the support of the Simons Foundation grant # Proofs of the Propositions 2.. Some preliminaries. Throughout this section will be a qc manifold of dimension + 3, ], with horizontal space H locally given as the kernel of a -form η = (η, η 2, η 3 ) with values in R 3, and Biquard connection with torsion T. Below we record some of the properties needed for this paper, see also 2] and 6] for a more expanded exposition. The Sp(n)Sp() structure on H is fixed by a positive definite symmetric tensor g and a rank-three bundle Q of endomorphisms of H locally generated by three almost complex structures I, I 2, I 3 on H satisfying the identities of the imaginary unit quaternions and also the conditions g(i s., I s.) = g(.,.) and 2g(I s X, Y ) = dη s (X, Y ). Associated with the Biquard connection is the vertical space V, which is complementary to H in T. In the case n = we shall make the usual assumption of existence of Reeb vector fields ξ, ξ 2, ξ 3, so that the connection is defined following D. Duchemin 5]. The fundamental 2-forms ω s of the fixed qc structure will be denoted by ω s, 2ω s H = dη s H, ξ ω s = 0, ξ V. In order to give some idea of the involved quantities we list a few more essential for us details. Recall that preserves the decomposition H V and the Sp(n)Sp() structure on H, g = 0, Γ(Q) Γ(Q) and its torsion on H is given by T (X, Y ) = X, Y ] V. Furthermore, for a vertical field ξ V, the endomorphism T ξ T (ξ,.) H of H belongs to the space (sp(n) sp()) gl() hence T (ξ, X, Y ) = g(t ξ X, Y ) is a well defined tensor field. The two Sp(n)Sp()-invariant trace-free symmetric 2-tensors T 0 (X, Y ) = g((t 0 ξ I + T 0 ξ 2 I 2 + T 0 ξ 3 I 3 )X, Y ), U(X, Y ) = g(ux, Y ) on H, introduced in 0], satisfy (2.) T 0 (X, Y ) + T 0 (I X, I Y ) + T 0 (I 2 X, I 2 Y ) + T 0 (I 3 X, I 3 Y ) = 0, U(X, Y ) = U(I X, I Y ) = U(I 2 X, I 2 Y ) = U(I 3 X, I 3 Y ). Note that when n =, the tensor U vanishes. The tensors T 0 and U determine completely the torsion endomorphism due to the identity 4, Proposition 2.3] which in view of (2.) implies 4T 0 (ξ s, I s X, Y ) = T 0 (X, Y ) T 0 (I s X, I s Y ), (2.2) T (ξ s, I s X, Y ) = T 0 (X, Y ) 3U(X, Y ). s=

5 ENTROPY CONVEXITY ON QC AND CR ANIFOLDS 5 The curvature of the Biquard connection is R =, ], ] with qc-ricci tensor and normalized qc-scalar curvature, defined by respectively by Ric(X, Y ) = a= g(r(e a, X)Y, e a ), 8n(n + 2)S = a= Ric(e a, e a ). According to 2] the Ricci tensor restricted to H is a symmetric tensor. Remarkably, the torsion tensor determines the qc-ricci tensor of the Biquard connection on in view of the formula, 0], (2.3) Ric(X, Y ) = (2n + 2)T 0 (X, Y ) + ( + 0)U(X, Y ) + S g(x, Y ). Finally, V ol η will denote the volume form, see 0, Chapter 8], (2.4) V ol η = η η 2 η 3 Ω n, where Ω = ω ω + ω 2 ω 2 + ω 3 ω 3 is the fundamental 4-form. We note the integration by parts formula (2.5) ( σ) V ol η = 0, where the (horizontal) divergence of a horizontal vector field σ Λ (H) is given by σ = tr H σ = σ(e a, e a ) for an orthonormal frame {e a } a= of the horizontal space Proof of Proposition.3. We start with a formula for the change of the dependent function in the P -function of f. To this effect, with f = f(f ), a short calculation shows the next identity 3 f(z, X, Y ) = f 3 F (Z, X, Y ) + f df (Z)dF (X)dF (Y ) Recalling definition (.5) we obtain + f 2 F (Z, X)dF (Y ) + f 2 F (Z, Y )df (X) + f 2 F (X, Y )df (Z). (2.6) P f (Z) = f P F (Z) + f F 2 df (Z) + 2f 2 F (Z, F ) + f ( F )df (Z) which implies the identity + f g( 2 F, ω s )df (I s Z), (2.7) P f ( f) = f 2 P F ( F ) + f f F 4 + 2f f 2 F ( F, F ) + f f F 2 F. In our case, since we are interested in expressing the integral of up f ( f) = e f P f ( f) in terms of the integral of a P -function of some function, equation (2.7) leads to the ordinary differential ( ) 2 equation u u u = const. Therefore, we let u = F 2 and find (2.8) up f ( f) = 4P F ( F ) + 8F 2 F 4 8F 2 F ( F, F ) 4F F 2 F. Now, the last three terms will be expressed back in the variable f which gives (2.9) up f ( f) = 4P F ( F ) + 4 f 4 + ] 2 f 2 f + 2 f( f, f) u At this point, we integrate the above identity and then apply the (integration by parts) divergence formula (2.5) in order to show 2 f( f, f)u V ol η = f 4 f 2 f ] u V ol η, 2 s=

6 6 D. VASSILEV which leads to (.6). The proof of Proposition.3 is complete Proof of Proposition.. The initial steps are identical to the Riemannian case 8] thus we skip the detailed computations and only sketch the common steps. Let w = 2 f f 2. Using the heat equation, exactly as in the Riemannian case, we have the identities (2.0) t f = f f 2, u f = u f 2 u, and u = ( f 2 f)u, which imply (2.) E (t) = and also ( t )(uw) V ol η (2.2) ( t )(uw) = 2g ( ( f), f) f 2] u. Next, we apply the qc Bochner formula 3, 2] 2 f 2 = 2 f 2 + g ( ( f), f) + 2(n + 2)S f 2 where Therefore, (2.3) + 2(n + 2)T 0 ( f, f) + 4(n + )U( f, f) + 4R f ( f), R f (Z) = 2 f(ξ s, I s Z). 2 ( t )(uw) = 2 f 2 2(n + 2)S f 2 2(n + 2)T 0 ( f, f) s= 4(n + )U( f, f) 4R f ( f) ] u. The next step is the computation of R f ( f)u V ol η in two ways as was done in 3, 2] for the Lichnerowicz type first eigenvalue lower bound but integrating with respect to V ol η rather than u V ol η as we need to do here. For ease of reading we will follow closely 7, Section 8..] but notice the opposite convention of the sub-laplacian in 7, Section 8..]. First with the help of the P -function, working similarly to 2, Lemma 3.2] where the integration was with respect to V ol η, we have (2.4) R f ( f)u V ol η = P f ( f) ( f)2 S f 2 + n + n U( f, f)]u V ol η + f 2 ( f)u V ol η, with the convention that in the case n = the formula is understood by formally dropping the term involving (the vanishing) tensor U. Notice the appearance of a new term in the last integral in comparison to the analogous formula in 7, Section 8.., p. 30]. Indeed, taking into account the Sp(n)Sp() invariance of R f ( f) and Ricci s identities we have, cf. 2, Lemma 3.2], hence (.5) gives R f (X) = s= a= 3 f(i s X, e a, I s e a ) + T 0 (X, f) 3U(X, f) ] ur f ( f) = P n( f) S f 2 + n + n U( f, f)] u + a= 3 f( f, e a, e a )u.

7 ENTROPY CONVEXITY ON QC AND CR ANIFOLDS 7 An integration by parts shows the validity of (2.4). On the other hand, we have (2.5) R f ( f)u V ol η = ] g( 2 f, ω s ) 2 + T 0 ( f, f) 3U( f, f) u V ol η, s= which other than using different volume forms is identical to the second formula in 7, Section 8.., p. 30]. Indeed, following 3, Lemma 3.4], using Ricci s identity 2 f(x, ξ s ) 2 f(ξ s, X) = T (ξ s, X, f) and (2.2), we have ( ) R f ( f) = 2 f(i s f, ξ s ) T 0 ( f, f) 3U( f, f) ] s= An integration by parts gives (2.5), noting that 3 s= df(ξ s)df(i s f) = 0 and taking into account that by Ricci s identity 2 f(x, Y ) 2 f(y, X) = 2 ω s (X, Y )df(ξ s ) we have g( 2 f, ω s )= a= 2 f(e a, I s e a ) = df(ξ s ). Now, working as in 7, Section 8.., p. 30], we subtract (2.5) and three times formula (2.4) from (2.3) which brings us to the following identity (2.6) 2 E (t) = ( 2 f) 0 2 2n + 2 s= L( f, f) ] u V ol η + 3Pf ( f) + 2( f) 2 3 f 2 f ] u V ol η, where ( 2 f) 0 2 is the square of the norm of the traceless part of the horizontal Hessian ( 2 f) 0 2 = 2 f 2 ( f) 2 + g( 2 f, ω s )] 2]. Next, we consider 2( f) 2 3 f 2 f ] u V ol η. Using the heat equation we have the identical to the Riemannian case relation, see (2.0), E (t) = d ( w u V ol η = 2( f) f 2 f f 4) u V ol η, dt hence (2.7) s= ( 2( f) 2 3 f 2 f ) u V ol η = d dt E(t) f 4 u V ol η. A substitution of the above formula in (2.6) gives 2n + d dt E(t) = ( 2 f) 0 2 2n + L( f, f) ] u V ol η + 3Pf ( f) f 4] u V ol η. 2 Finally, we invoke Proposition.3 in order to complete the proof.

8 8 D. VASSILEV 3. The CR case In this section, following the method we employed in the qc case, we prove the monotonicity formula in the CR case stated in Proposition.4. This implies the monotonicity of the entropy like energy which was proved earlier in 4]. Throughout the section will be a (2n + )-dimensional strictly pseudoconvex (integrable) CR manifold with a fixed pseudohermitian structure defined by a contact form η and a complex structure J on the horizontal space H = Ker η. The fundamental 2-form is defined by ω = 2 dη and the Webster metric is g(x, Y ) = ω(jx, Y ) which is extended to a Riemannian metric on by declaring that the Reeb vector field associated to η is of length one and orthonormal to the horizontal space. We shall denote by the associated Tanaka-Webster connection 9] and 20, 2], while u = tr g ( 2 u) will be the negative sub-laplacian with the trace taken with respect to an orthonormal basis of the horizontal 2n-dimensional space. Finally, we define the Ricci type tensor (3.) L(X, Y ) = ρ(jx, Y ) + 2nA(JX, Y ) recalling that on a CR manifold we have (3.2) Ric(X, Y ) = ρ(jx, Y ) + 2(n )A(JX, Y ), where ρ is the (, )-part of the pseudohermitian Ricci tensor (the Webster Ricci tensor) while the (2, 0) + (0, 2)-part is the Webster torsion A, see 6, Chapter 7] for the expressions in real coordinates of these known formulas 20, 2], see also 6]. With the above convention in place, as in 4], for a positive solution of (.) we consider the entropy (.2) and energy (.3), where (3.3) V ol η = η (dη) 2n. We turn to the proof of Proposition.4. For a function f we define the one form, (3.4) P f (X) = 3 f(x, e b, e b ) + 3 f(jx, e b, Je b ) + A(X, J f) so that the fourth order CR-Paneitz operator is given by (3.5) C(f) = P = ( ea P )(e a ) = 4 f(e a, e a, e b, e b ) + 4 f(e a, Je a, e b, Je b ) A(J f) g( 2 f, JA). By 7], when n > a function f C 3 () satisfies the equation Cf = 0 iff f is CR-pluriharmonic. Furthermore, the CR-Paneitz operator is non-negative, f Cf V ol η = P f ( f) V ol η 0. On the other hand, in the three dimensional case the positivity condition is a CR invariant since it is independent of the choice of the contact form by the conformal invariance of C proven in 9]. The non-negativity of the CR-Paneitz operator is relevant in the embedding problem for a three dimensional strictly pseudoconvex CR manifold. As shown in 3] if the pseudohermitian scalar curvature of is positive and C is non-negative, then is embeddable in some C n We turn to the proof of Proposition.4. Taking into account (2.2) and the CR Bochner formula 8], (3.6) 2 f 2 = 2 f 2 + g( ( f), f) + Ric( f, f) + 2A(J f, f) + 4R f ( f),

9 ENTROPY CONVEXITY ON QC AND CR ANIFOLDS 9 where R f (Z) = df(ξ, JZ), see 7, Section 7.] and references therein but note the opposite sign of the sub-laplacian, we obtain the next identity (3.7) 2 ( t )(uw) = 2 f 2 Ric( f, f) 2A( f, f) 4R f ( f) ] u. Since (2.) still holds, working as in the qc case we compute R F ( f)u V ol η in two ways 8, Lemma 4] and 5, Lemma 8.7] following the exposition 7]. From Ricci s identity it follows df(ξ) = 2n g( 2 f, ω). Hence 2 f(x, Y ) 2 f(y, X) = 2ω(X, Y )df(ξ) 2 f(jz, ξ) = 2n 2n b= 3 f(jz, e b, Je b ), where {e b } 2n b= is an orthonormal basis of the horizontal space. Applying Ricci s identity it follow (3.8) R f (Z) = 2 f(ξ, JZ) = 2n Taking into account (3.4), the last formula gives 2 f(x, ξ) 2 f(ξ, X) = A(X, f) 2n b= R f (Z) = 2n P f (Z) + A(JZ, f) + 2n 3 f(jz, e b, Je b ) A(JZ, f). 2n b= 3 f(z, e b, e b ). Now, an integration by parts shows the next identity (3.9) R f ( f)u V ol η = 2n P f ( f) + A(J f, f) 2n ( f)2 + 2n f 2 ( f) ] u V ol η. On the other hand, using again (3.8) but now we integrate and then use integration by parts, we have (3.0) R f ( f)u V ol η = 2n g( 2 f, ω) 2 A(J f, f) ] u V ol η. At this point, exactly as in the qc case, we subtract (3.0) and three times formula (3.9) from (3.7), which gives E (t) = ( 2 f) L( f, f) ] u V ol η + 3Pf ( f) + 2( f) 2 3 f 2 f ] u V ol η, 2n where ( 2 f) 0 2 is the square of the norm of the traceless part of the horizontal Hessian ( 2 f) 0 2 = 2 f 2 ( f) 2 + g( 2 f, ω) 2]. 2n Taking into account that the formulas in Proposition.3 and (2.7) hold unchanged we complete the proof.

10 0 D. VASSILEV References ] Biquard, O., Quaternionic contact structures, Quaternionic structures in mathematics and physics (Rome, 999), (electronic), Univ. Studi Roma La Sapienza, Roma, ] Biquard, O., étriques d Einstein asymptotiquement symétriques, Astérisque 265 (2000). 4, 5 3] Chanillo, S., Chiu, H.-L., & Yang, P., Embeddability for 3-dimensional Cauchy-Riemann manifolds and CR Yamabe invariants. Duke ath. J. 6 (202), no. 5, ] Chang, Shu-Cheng; Wu, Chin-Tung, The entropy formulas for the CR heat equation and their applications on pseudohermitian (2n+)-manifolds. Pacific J. ath. 246 (200), no., 29., 3, 4, 8 5] Duchemin, D., Quaternionic contact structures in dimension 7, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, ] Dragomir, S. & Tomassini, G. Differential geometry and analisys on CR manifolds, Progress in ath. vol. 246, Birkhäuser Boston, Inc., Boston, A, ] Graham,C.R., & Lee, J.., Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains. Duke ath. J., 57(988), ] Greenleaf, A., The first eigenvalue of a sublaplacian on a pseudohermitian manifold. Commun. Partial Diff. Equations, 0 (985), no. 2, , 9 9] Hirachi, K., Scalar pseudo-hermitian invariants and the Szegö kernel on three-dimensional CR manifolds, in: Complex Geometry, 990 Osaka Conf. Proc. arcel Dekker Lect. Notes Pure Appl. ath. 43 (993), ] Ivanov, S., inchev, I., & Vassilev, D., Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem, emoirs Amer. ath. Soc. vol. 23, number 086; 3, 4, 5 ] Ivanov, S. & Petkov, A., A heat equation on a quaternionic contact manifold, arxiv: , 4 2] Ivanov, S., Petkov, A., & Vassilev, D., The sharp lower bound of the first eigenvalue of the sub-laplacian on a quaternionic contact manifold in dimension 7, Nonlinear Anal-Theory 93 (203) , 3, 6 3] Ivanov, S., Petkov, A., & Vassilev, D., The sharp lower bound of the first eigenvalue of the sub-laplacian on a quaternionic contact manifold, J. Geom. Anal. 24 (204), no. 2, , 6, 7 4] Ivanov, S., & Vassilev, D., Conformal quaternionic contact curvature and the local sphere theorem, J. ath. Pures Appl. 93 (200), ], An Obata type result for the first eigenvalue of the sub-laplacian on a CR manifold with a divergence free torsion, J. Geom. 03 (202), ] Ivanov, S., & Vassilev, D., Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 20. 4, 8 7] Ivanov, S., & Vassilev, D., The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds, Nonlinear Analysis 26 (205) , 3, 6, 7, 9 8] Ni, L., The entropy formula for linear heat equation. J. Geom. Anal. 4 (2004), no., , 3, 6 9] Tanaka, N., A differential geometric study on strongly pseudo-convex manifolds, Lectures in athematics, Department of athematics, Kyoto University, No. 9. Kinokuniya Book-Store Co., Ltd., Tokyo, ] Webster, S.., Real hypersurfaces in complex space, Thesis, University of California, ] Webster, S.., Pseudo-hermitian structures on a real hypersurface, J.Diff. Geom., 3 (979), (Dimiter Vassilev) Department of athematics and Statistics, University of New exico, Albuquerque, New exico, address: vassilev@math.unm.edu