EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6985 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Cheeger-Type Inequality for the Sub-Laplacian on Pseudo-Hermitian CR Manifolds Ali Ben Ahmed Department of Mathematics, University College Al-Khafji, University of Hafr Al Batin, Saudi Arabia Abstract. In this paper, we introduce a CR Cheeger constant and establish Cheeger-type and Buser-type inequalities for the first nonzero eigenvalue of the sub-Laplacian on compact strictly pseudoconvex pseudo-Hermitian CR manifolds. These results extend classical isoperimetric bounds from Riemannian to CR geometry. Applications are given to model examples, including the Heisenberg quotients, the standard CR sphere and Rossi’s non-embeddable deformations. 2020 Mathematics Subject Classifications: 32V05, 53C17, 58J50 Key Words and Phrases: CR manifolds, sub-Laplacian, Cheeger inequality, isoperimetric constant 1. Introduction The discovery of Cheeger’s inequality in 1970 [1] marked a turning point in geometric analysis, establishing a precise link between isoperimetric geometry and spectral theory. For a compact Riemannian manifold (M, g) without boundary, Cheeger showed that the first nonzero eigenvalue λ1(∆g) of the Laplace–Beltrami operator ∆g satisfies λ1(∆g) ≥ 1 4 h(M)2, (1.1) where h(M) is the isoperimetric constant. This inequality has had far-reaching conse- quences, providing a bridge between geometry and analysis through applications to Sobolev inequalities, heat kernel bounds and concentration phenomena. A decade later, Buser [2] complemented Cheeger’s estimate with an upper bound depending on curvature, thereby giving a two-sided description of the spectral gap in terms of isoperimetry. The present work develops this classical picture in the strictly pseudoconvex CR manifolds. Here the natural second-order operator is the horizontal sub-Laplacian ∆b, acting along the contact distribution determined by a pseudo-Hermitian structure. Extending Cheeger’s method from the elliptic to the hypoelliptic world presents several difficulties: DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6985 Email address: benahmedal@gmail.com (A. Ben Ahmed) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 2 of 16 the horizontal bundle exhibits anisotropic scaling, notions of perimeter must be recast using horizontal BV theory and torsion terms arising from the Tanaka–Webster connection affect the analytic inequalities. These challenges motivated the development of horizontal analysis in Carnot–Carathéodory spaces, most notably by Folland and Stein [3], Jerison and Lee in their study of the CR Yamabe problem [4, 5] and Franchi–Serapioni–Serra Cassano and Garofalo–Nhieu [6, 7], who introduced horizontal BV functions, co-area formulas and isoperimetric control. More recently, curvature-dimension methods of Baudoin and Garofalo [8] and sharp inequalities of Frank and Lieb [9] have advanced the analytic framework in Heisenberg-type geometries. Against this background, we establish a CR analogue of Cheeger’s inequality. We define a natural CR Cheeger constant hCR(M) in terms of horizontal perimeter and contact volume and prove that the first positive eigenvalue of ∆b satisfies λ1(∆b) ≥ C∗ hCR(M)2, where C∗ depends only on the homogeneous dimension and on controlled pseudo-Hermitian quantities such as curvature and torsion. The proof adapts Cheeger’s variational co- area method using the horizontal BV framework of [6, 7, 10], together with sharp local isoperimetric comparisons. In symmetric model spaces, including compact quotients of the Heisenberg group and the standard CR sphere, the optimal constant C∗ = 1 4 is recovered, confirming sharpness in these cases. In addition, inspired by Buser’s 1982 argument, we establish a conditional Buser–CR inequality showing that λ1(∆b) ≲ hCR(M)2 + hCR(M), provided the metric measure space (M,dCC , µ) satisfies standard sub-Riemannian analytic assumptions, namely volume doubling, a (1, 1)-Poincaré inequality and Gaussian heat kernel bounds, as in the works of Jerison–Sánchez-Calle [11] and Baudoin–Garofalo [8]. These conditions are verified in many natural examples, including Sasakian manifolds and Heisenberg-type groups and show that the two-sided spectral control familiar from the Riemannian theory extends, in an appropriate form, to the CR framework. Finally, to illustrate the scope of the results we provide explicit computations in model geometries: compact Heisenberg quotients, the standard CR sphere S2n+1 and Rossi’s non-embeddable deformations of S3. These examples highlight not only the sharpness of the constants but also the role of torsion and non-embeddability in influencing the spectral gap. In this way, the CR Cheeger and Buser inequalities we establish extend a classical Riemannian paradigm into the hypoelliptic setting of CR geometry, enriching the broader landscape of CR spectral theory. 2. Preliminaries and analytic tools 2.1. Pseudo-Hermitian CR structures Let M be a smooth, connected manifold. A strictly pseudoconvex pseudo-Hermitian structure on M is specified by a contact 1-form θ and an almost complex structure J on A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 3 of 16 the contact bundle H = ker θ such that the Levi form Lθ(X,Y ) := dθ(X,JY ), X, Y ∈ Γ(H), is positive definite. The Reeb vector field T satisfies θ(T ) = 1, dθ(T, ·) = 0. The natural volume form is µ := θ ∧ (dθ)n, (2.1) and the homogeneous dimension is Q = 2n+ 2. Given a smooth function u, its horizontal gradient ∇bu is the unique horizontal vector field satisfying du(X) = Lθ(∇bu,X) for all X ∈ Γ(H). The horizontal divergence divH computed with respect to µ yields the (positive) sub-Laplacian ∆bu = − divH(∇bu), (2.2) which is essentially self-adjoint on L2(M,µ) with discrete spectrum on compact M (see [12]). 2.2. Horizontal perimeter and BV functions A key analytic tool in extending isoperimetric methods to the CR setting is the notion of perimeter adapted to the horizontal distribution. This is provided by horizontal bounded variation (BVH) theory, developed by Franchi–Serapioni–Serra Cassano [6] and further refined in [7]. In this subsection we recall the necessary framework. Horizontal variation. Let u ∈ L1(M), where M is a compact strictly pseudoconvex pseudo-Hermitian CR manifold with contact volume µ = θ ∧ (dθ)n. The horizontal total variation of u is defined by |Du|H(M) = sup {∫ M u divµ φ dµ : φ ∈ C1 c (M ;H), ∥φ∥∞ ≤ 1 } , (2.3) where divµ is the horizontal divergence with respect to µ. We say u ∈ BVH(M) if |Du|H(M) <∞. Horizontal perimeter. For a measurable set E ⊂M with finite horizontal variation of its indicator function, the horizontal perimeter of E in M is defined by PerH(E;M) := |D1E |H(M). This generalizes the Riemannian notion of perimeter to the CR framework. Proposition 1 (Horizontal co-area formula; [6, 7]). If u ∈ BVH(M), then |Du|H(M) = ∫ +∞ −∞ PerH({u > t};M) dt. (2.4) Moreover, if u ∈ Lip(M), then |Du|H(M) = ∫ M |∇bu| dµ. (2.5) A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 4 of 16 Proof. We present a systematic argument in three steps. Step 1: The smooth case. Suppose u ∈ C∞(M). For a regular value t, let Et = {u > t}. The level set {u = t} is a smooth hypersurface with Riemannian unit normal N = ∇u/|∇u|. Writing NH for the projection of N onto the horizontal bundle H, one has |NH | = |∇bu|/|∇u|. By the characterization of horizontal perimeter for smooth sets [6], PerH(Et;M) = ∫ {u=t} |NH | dσt = ∫ {u=t} |∇bu| |∇u| dσt, where dσt is the induced surface measure. Applying the classical Riemannian co-area formula to the integrand ψ(x) = |∇bu(x)|/|∇u(x)| yields∫ M |∇bu| dµ = ∫ +∞ −∞ PerH(Et;M) dt. This proves both (2.4) and (2.5) for smooth u. Step 2: Extension to BVH . Let u ∈ BVH(M). By the approximation theorem in horizontal BV theory [6, Thm. 3.1], there exists a sequence uk ∈ C∞(M) such that uk → u in L1(M) and |Duk|H(M) → |Du|H(M). From Step 1, |Duk|H(M) = ∫ +∞ −∞ PerH({uk > t};M) dt. By convergence in measure, 1{uk>t} → 1{u>t} for a.e. t and by lower semicontinuity of perimeter [6, Thm. 4.2], PerH({u > t};M) ≤ lim inf k→∞ PerH({uk > t};M). Integrating and applying Fatou’s lemma gives∫ +∞ −∞ PerH({u > t};M) dt ≤ |Du|H(M). The reverse inequality follows by the layer-cake representation of u combined with the definition of perimeter (see [7]). Thus equality holds, proving (2.4). Step 3: Lipschitz functions. If u ∈ Lip(M), then ∇bu exists µ-a.e. by Rademacher’s theorem in Carnot–Carathéodory spaces [7]. Using (2.3) and integration by parts, one finds |Du|H(M) ≤ ∫ M |∇bu| dµ. The reverse inequality is obtained by testing with vector fields aligned with −∇bu/|∇bu| on the set where |∇bu| > ε and letting ε→ 0. Hence (2.5) holds. The proof is complete. A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 5 of 16 2.3. Isoperimetric and functional inequalities We will rely on isoperimetric and Poincaré inequalities available in strictly pseudoconvex CR manifolds under mild quantitative controls (volume doubling, CC-diameter bounds, or curvature-dimension assumptions). We state them in a form convenient for our spectral estimates; precise constants depend on the chosen normalization and on geometric bounds (torsion, lower horizontal Ricci). Proposition 2 (Horizontal isoperimetric inequality). Let (M, θ, J) be a compact strictly pseudoconvex pseudo-Hermitian CR manifold and let Q = 2n+ 2 denote its homogeneous dimension. Then there exists a constant Ciso > 0, depending only on Q and quantitative pseudo-Hermitian bounds, such that for every measurable set E ⊂M with µ(E) ≤ 1 2µ(M) one has µ(E) Q−1 Q ≤ Ciso PerH(E;M), (2.6) where µ is the pseudo-Hermitian volume and PerH(E;M) the horizontal perimeter. Proof. The inequality is the CR analogue of the classical isoperimetric inequality in Euclidean and Carnot–Carathéodory geometries. We proceed in three steps. Step 1: Model case on the Heisenberg group. On the Heisenberg group Hn with its standard CR structure, the sharp isoperimetric inequality is known: |E| Q−1 Q ≤ C PerH(E), ∀E ⊂ Hn, where | · | denotes Haar measure and PerH the horizontal perimeter. This result goes back to Pansu and was rigorously established in [7, 9]. The sharp constant C depends only on n. Step 2: Local comparison on CR manifolds. In a strictly pseudoconvex CR manifold (M, θ, J), privileged coordinates (in the sense of Rothschild–Stein) allow comparison between small Carnot–Carathéodory (CC) balls in M and balls in Hn. More precisely, under uniform bounds on the pseudo-Hermitian structure (torsion, curvature and injectivity radius), small CC balls are quantitatively close—in both metric and measure—to Heisenberg balls. Consequently, relative isoperimetric inequalities valid in Hn transfer locally to M with controlled constants (see [6, 7]). Step 3: Globalization. Cover M by finitely many CC balls of small radius. The relative isoperimetric inequality in each ball, together with a partition-of-unity and a standard compactness argument, yields a global inequality of the form min{µ(E), µ(M \ E)} Q−1 Q ≤ Ciso PerH(E;M), valid for all measurable E ⊂M . Restricting to µ(E) ≤ 1 2µ(M) gives (2.6). Thus the horizontal isoperimetric inequality (2.6) holds with a constant Ciso depending only on Q and on quantitative bounds of the pseudo-Hermitian structure. A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 6 of 16 Proposition 3 (Horizontal Poincaré inequality). Let (M,dCC , µ) be a compact pseudo- Hermitian manifold with bounded Carnot–Carathéodory diameter diamCC(M) <∞, and assume that M satisfies the local doubling property and a local Poincaré inequality. Then there exists a constant CP > 0 such that for every Lipschitz function u with vanishing average ∫ M u dµ = 0, one has∫ M |u| dµ ≤ CP diamCC(M) ∫ M |∇bu| dµ. (2.7) Proof. Standard from the (1,1)-Poincaré inequality in the Carnot–Carathéodory setting and the fact that M is compact; see [4, 7, 13]. Remark 1. All constants Ciso, CP can be made explicit once one fixes quantitative bounds on torsion, horizontal Ricci-like quantities (via curvature-dimension inequalities) and the Carnot–Carathéodory diameter. Later we will express the dependence of spectral constants on these geometric data. 3. The CR Cheeger constant and main theorem Definition 1 (CR Cheeger constant). Let (M, θ, J) be compact strictly pseudoconvex CR manifold. The CR Cheeger constant is hCR(M) := inf E⊂M PerH(E;M) min{µ(E), µ(M \ E)} , (3.1) infimum taken over sets of finite horizontal perimeter. Our main spectral result is the following. Theorem 1 (Cheeger–CR inequality). Let (M, θ, J) be a compact strictly pseudoconvex pseudo-Hermitian CR manifold. Denote by λ1 = λ1(∆b) the first positive eigenvalue of the sub-Laplacian ∆b on functions (with respect to the measure µ). Then λ1 ≥ C∗(Q,T) hCR(M)2, (3.2) where Q = 2n+ 2 and C∗(Q,T) ∈ (0, 14 ] is an explicit constant depending only on Q and controlled pseudo-Hermitian bounds T. In torsion-free model settings (compact Heisenberg quotients, standard CR sphere) one can take C∗ = 1 4 under the usual normalization of ∆b. Proof of Theorem 1 The following propositions provide the key technical ingredients for the proof of the main theorem. A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 7 of 16 Proposition 4 ( Cheeger slicing inequality, horizontal version). Let f ∈ Lip(M) with∫ M f dµ = 0. Then∫ M |∇bf | dµ ≥ hCR(M) ∫ +∞ −∞ min{µ({f > t}), µ({f ≤ t})} dt. (3.3) Consequently, there exists t0 ∈ R such that∫ M |∇bf | dµ ≥ 1 2 hCR(M) ∫ M |f | dµ. (3.4) Proof. Apply the horizontal co-area formula (Proposition 1) to f :∫ M |∇bf | dµ = ∫ ∞ −∞ PerH({f > t};M) dt. By definition of hCR(M), PerH({f > t};M) ≥ hCR(M) min{µ({f > t}), µ({f ≤ t})}. Integrate in t to obtain (3.3). The identity∫ ∞ −∞ min{µ({f > t}), µ({f ≤ t})} dt = 1 2 ∫ M |f | dµ follows from the Cavalieri representation applied to |f |; combining yields (3.4). Proposition 5 (From L1 control to L2 estimate). Let f ∈ Lip(M) with ∫ M f dµ = 0. Then ∫ M |∇bf |2 dµ ≥ hCR(M)2 4C2 PdiamCC(M)2 ∫ M f2 dµ, (3.5) where CP and diamCC(M) are as in Proposition 3. In particular, if one normalizes so that CPdiamCC(M) = 1 then the prefactor is hCR(M)2/4. Proof. From Proposition 4 we have∫ M |∇bf | dµ ≥ 1 2hCR(M) ∫ M |f | dµ. By the Cauchy–Schwarz inequality,∫ M |∇bf | dµ ≤ (∫ M |∇bf |2 dµ )1/2 µ(M)1/2, and by Proposition 3 (Poincaré with zero mean)∫ M |f | dµ ≤ CP diamCC(M) ∫ M |∇bf | dµ. A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 8 of 16 Combining and eliminating ∫ M |∇bf | dµ yields(∫ M |∇bf |2 dµ )1/2 ≥ hCR(M) 2CPdiamCC(M) (∫ M f2 dµ )1/2 , and squaring gives (3.5). By the Rayleigh quotient characterization, λ1 = inf f∈C∞(M)∫ f=0 ∫ M |∇bf |2 dµ∫ M f2 dµ . Applying Proposition 5 to any admissible f yields the claimed lower bound with C∗(Q,T) = 1 4C2 PdiamCC(M)2 , where the dependence on Q and T arises through the Poincaré constant CP and the diameter control (both of which can be quantified under curvature-dimension or torsion bounds). In symmetric torsion-free models one may arrange the normalization so that CPdiamCC(M) = 1, recovering the classical 1/4 factor. This completes the proof of Theorem 1. Remark 2. The chain of inequalities shows precisely where pseudo-Hermitian geometry enters: the co-area formula is horizontal, the isoperimetric profile defines hCR and the Poincaré constant depends on doubling and curvature-dimension assumptions that may involve Webster torsion. Hence explicit dependence of C∗ on torsion can be tracked by quantifying CP . 4. A conditional Buser–CR upper bound While Cheeger-type inequalities provide lower bounds for the first eigenvalue of the horizontal sub-Laplacian, a converse bound requires stronger analytic controls. In this section we establish a CR analogue of Buser’s inequality, conditional on standard sub- Riemannian analytic hypotheses. The proof adapts Buser’s variational strategy to the horizontal distribution. Theorem 2 (Buser–CR inequality (conditional)). Let (M, θ, J) be a compact strictly pseudoconvex pseudo-Hermitian CR manifold and let ∆b denote its horizontal sub-Laplacian. Assume: (A1) (M,dCC , µ) satisfies a volume doubling property and a (1, 1)-Poincaré inequality with uniform constants; (A2) the heat kernel pt(x, y) of ∆b satisfies Gaussian upper bounds: there exist A,B > 0 such that pt(x, y) ≤ A µ(B(x, √ t)) exp ( − dCC(x, y) 2 Bt ) , ∀ 0 < t ≤ 1. (4.1) A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 9 of 16 Then there exist constants C1, C2 > 0, depending only on the doubling, Poincaré and heat-kernel constants, such that λ1(∆b) ≤ C1 hCR(M)2 + C2 hCR(M), (4.2) where hCR(M) is the CR Cheeger constant. Proof of Theorem 2 We follow Buser’s original strategy, adapting each step to the CR setting. Step 1: Choice of an almost minimizer. By definition of hCR(M) there exists a measurable set E ⊂M of finite horizontal perimeter such that PerH(E;M) min{µ(E), µ(M \ E)} ≤ 2hCR(M). (4.3) We may assume µ(E) ≤ 1 2µ(M) by symmetry. By standard regularization arguments (see [6]), E can be approximated by open sets with smooth horizontal boundary without altering the ratio in (4.3). Step 2: Cutoff function construction. Fix r > 0 small. Let Er = {x ∈ M : dCC(x,E) < r} denote the Carnot–Carathéodory r-neighborhood. Choose a Lipschitz cutoff ϕ ∈ Lip(M) such that ϕ ≡ 1 on E, ϕ ≡ 0 on M \ Er, |∇bϕ| ≤ C r a.e., where, C is a universal constant depending only on the construction. Such a function can be obtained via horizontal mollification and partition of unity. Proposition 6 (Rayleigh quotient estimate). Let ϕ be as above. Then R(ϕ) := ∫ M |∇bϕ|2 dµ∫ M (ϕ− ϕ)2 dµ ≤ C3 r2 + C4 r PerH(E;M) µ(E) , (4.4) for constants C3, C4 depending only on the doubling and Poincaré data. Proof. We estimate numerator and denominator separately. Numerator. By the gradient bound |∇bϕ| ≤ C/r and support properties,∫ M |∇bϕ|2 dµ ≤ C2 r2 µ(Er \ E). By the horizontal co-area formula (Proposition 1), µ(Er \ E) ≤ C r PerH(E;M), A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 10 of 16 for a universal constant C depending only on doubling. Hence∫ M |∇bϕ|2 dµ ≤ C ′ r PerH(E;M). (4.5) Denominator. Since ϕ equals 1 on E and vanishes outside Er, its average satisfies ϕ = 1 µ(M) ∫ M ϕ dµ ≤ µ(Er) µ(M) . Thus ∫ M (ϕ− ϕ)2 dµ ≥ ∫ E (1− ϕ)2 dµ ≥ (1− µ(Er) µ(M) ) 2µ(E). By doubling, µ(Er) ≤ C µ(E) when r is chosen small compared to diamCC(M). Hence∫ M (ϕ− ϕ)2 dµ ≥ c µ(E), (4.6) for some c > 0 depending only on doubling. Rayleigh quotient. Combining (4.5) and (4.6) yields R(ϕ) ≤ C ′′ r PerH(E;M) µ(E) . Finally, an additional term O(r−2) arises from the contribution of µ(Er \ E) when E is small compared to M , leading to the full bound (4.4). Step 3: Optimization. By (4.3), PerH(E;M)/µ(E) ≤ 2hCR(M). Proposition 6 gives R(ϕ) ≤ C3 r2 + 2C4 r hCR(M). Optimizing in r by taking r ∼ 1/hCR(M) yields R(ϕ) ≤ C1 hCR(M)2 + C2 hCR(M). Step 4: Variational principle. Since ϕ is non-constant, the Rayleigh quotient R(ϕ) bounds λ1(∆b) from above: λ1(∆b) ≤ R(ϕ). This proves Theorem 2. . Remark 3. The theorem is conditional on assumptions (A1)–(A2). These conditions are satisfied in many natural examples, such as compact quotients of Heisenberg-type groups and compact Sasakian manifolds with uniformly bounded Webster curvature and torsion. The dependence on torsion is only through the analytic constants controlling the doubling and Poincaré properties, rather than explicitly in the inequality (4.2). A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 11 of 16 5. Model computations and examples 5.1. Compact quotients of the Heisenberg group Let Hn denote the (2n+1)-dimensional Heisenberg group with its standard left-invariant contact form and horizontal distribution. For a compact quotient M = Γ\Hn (with Haar measure), the horizontal structure is homogeneous and torsion-free; classical isoperimetric and Sobolev inequalities are sharp in this case [3, 9]. Hence the Cheeger–CR inequality holds with C∗ = 1 4 under the standard normalization of ∆b on M . 5.2. The standard CR sphere S2n+1 The standard CR sphere S2n+1 ⊂ Cn+1 with contact form induced by the Euclidean form is the canonical compact model. Important spectral facts (Folland–Stein [3]) indicate that eigenvalues of the sub-Laplacian acting on functions arise from spherical harmonics and with a convenient normalization, are of the form λk = k(k + 2n), k ∈ N. Thus the first positive eigenvalue corresponds to k = 1, giving λ1 = 2n + 1. (Different normalizations of θ or ∆b shift these numbers; the stated relation matches the standard choice in [3].) By symmetry, isoperimetric minimizers are spherical caps; hence hCR(S 2n+1) is a positive constant depending only on n and the Cheeger–CR lower bound is consistent with these values. 6. Rossi’s spheres and spectral perturbation 6.1. Historical and mathematical context In 1965 Rossi [14] produced one of the first explicit examples of compact strictly pseudoconvex CR manifolds that are not globally embeddable as hypersurfaces in any CN . Rossi’s construction demonstrated that integrability and embeddability in CR geometry are subtler than in the almost-complex case and triggered many subsequent works (Kohn, Nirenberg, Burns–Epstein, Huang–Siu and others). Rossi’s deformations are defined on the underlying C∞ manifold S3 and yield inequiva- lent CR structures with different analytic properties. 6.2. An explicit local deformation We present a local description of a family of deformations which will be useful for discussing spectral perturbation. Let S3 ⊂ C2 with coordinates (z1, z2) and let L = z2 ∂ ∂z1 − z1 ∂ ∂z2 (6.1) A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 12 of 16 be the standard CR (1, 0) vector field. Rossi’s idea is to perturb the CR structure by adding a small non-holomorphic component; one convenient model family is Lε = L + εΨ(z, z̄) ∂ ∂zj , (6.2) where Ψ is a suitable smooth function and ε a small real parameter; the precise choice of Ψ must ensure strict pseudoconvexity is preserved for small ε. A commonly used simplified form is Lλ = z2 ∂ ∂z1 − z1 ∂ ∂z2 + λ ( z1 ∂ ∂z1 + z2 ∂ ∂z2 ) , λ ∈ R, (6.3) which for λ ≠ 0 gives an inequivalent CR structure for small |λ|; see [14–16] for constructions and non-embeddability proofs. 6.3. Implications for spectral geometry The Rossi family provides a natural family to test stability of the Cheeger–CR inequality and the sensitivity of λ1(∆b) to torsion and non-embeddability. Two complementary questions arise: (i) Stability: how does λ1(∆ λ b ) vary with the deformation parameter λ? In particular, does the Cheeger–CR lower bound remain uniform in λ for small deformations? (ii) Detectability: can spectral data (e.g. λ1, heat trace asymptotics) detect embeddability or torsion in the CR structure? Proposition 7 (First-order spectral perturbation for simple eigenvalues). Let {∆λ b }λ∈(−λ0,λ0) be a C1-family of self-adjoint realizations of the sub-Laplacian on a fixed compact manifold M , acting on the fixed Hilbert space L2(M,µ), where µ is a fixed smooth reference measure. Assume: (i) each ∆λ b has compact resolvent on L2(M,µ); (ii) the dependence λ 7→ ∆λ b is C1 in the sense of graph-norm bounded operators on a common dense domain D (see Remark 4 below); (iii) λ1(0) is a simple eigenvalue with normalized real-valued eigenfunction ϕ0 ∈ D, ∥ϕ0∥L2(µ) = 1. Then there exist C1 maps λ 7→ λ1(λ) and λ 7→ ϕλ ∈ D with ∆λ bϕλ = λ1(λ)ϕλ, ∥ϕλ∥L2(µ) = 1, ϕ0 = ϕλ ∣∣ λ=0 and the first derivative at λ = 0 satisfies the Hellmann–Feynman identity d dλ ∣∣∣ λ=0 λ1(λ) = 〈 ∆̇0 bϕ0, ϕ0 〉 L2(M,µ) , (6.4) where ∆̇0 b := d dλ ∣∣ λ=0 ∆λ b exists as a symmetric operator on D. A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 13 of 16 Proof. Step 1: Simple eigenvalue branch and differentiability. By (i) the spectrum of ∆λ b is pure point with finite multiplicities accumulating only at +∞. By (ii) and classical Kato theory (see [17, Ch. II, Thm. 5.8; Ch. VII, §3]), the isolated simple eigenvalue λ1(0) admits a unique C1 continuation λ 7→ λ1(λ) and a C1 choice of normalized eigenvectors λ 7→ ϕλ ∈ D satisfying ∆λ bϕλ = λ1(λ)ϕλ, ∥ϕλ∥L2(µ) = 1. (6.5) Step 2: Gauge choice. Differentiability of ϕλ is not unique up to a λ-dependent phase. Fix the parallel transport gauge ⟨ϕ̇λ, ϕλ⟩L2(µ) = 0 for all λ, (6.6) where dot denotes d dλ . This can always be achieved by multiplying ϕλ by a suitable C1 real phase factor. In particular at λ = 0 we have ⟨ϕ̇0, ϕ0⟩ = 0. (6.7) Step 3: Differentiate the eigenvalue equation. Differentiate (6.5) at λ = 0 in L2(M,µ): ∆̇0 b ϕ0 + ∆0 b ϕ̇0 = λ̇1(0)ϕ0 + λ1(0) ϕ̇0. Take the L2(µ) inner product with ϕ0 and use self-adjointness of ∆0 b :〈 ∆̇0 b ϕ0, ϕ0 〉 + 〈 ∆0 b ϕ̇0, ϕ0 〉 = λ̇1(0) ⟨ϕ0, ϕ0⟩︸ ︷︷ ︸ =1 + λ1(0) ⟨ϕ̇0, ϕ0⟩. Since ∆0 b is self-adjoint and ∆0 bϕ0 = λ1(0)ϕ0,〈 ∆0 b ϕ̇0, ϕ0 〉 = 〈 ϕ̇0, ∆ 0 bϕ0 〉 = λ1(0) ⟨ϕ̇0, ϕ0⟩. By the gauge condition (6.7) these terms cancel. Therefore〈 ∆̇0 b ϕ0, ϕ0 〉 = λ̇1(0), which is precisely (6.4). Step 4: Alternative derivation via Rayleigh quotient. For completeness, we give a form-theoretic argument which is often convenient in CR geometry. Assume each ∆λ b is associated with a symmetric closed quadratic form aλ on a common dense form domain V (e.g. V =W 1,2 H (M)) and that λ 7→ aλ is C1 in the sense ȧ0(u, v) := d dλ ∣∣∣ λ=0 aλ(u, v) exists for all u, v ∈ V. For the normalized eigenbranch (λ1(λ), ϕλ) one has the Rayleigh identity λ1(λ) = aλ(ϕλ, ϕλ), ∥ϕλ∥L2(µ) = 1. A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 14 of 16 Differentiate at λ = 0: λ̇1(0) = ȧ0(ϕ0, ϕ0) + 2 a0(ϕ0, ϕ̇0). But a0(ϕ0, ·) represents the bounded functional v 7→ ⟨∆0 bϕ0, v⟩ = λ1(0)⟨ϕ0, v⟩, hence a0(ϕ0, ϕ̇0) = λ1(0)⟨ϕ0, ϕ̇0⟩ = 0 by (6.7). Therefore λ̇1(0) = ȧ0(ϕ0, ϕ0). Since ȧ0(ϕ0, ϕ0) = ⟨∆̇0 bϕ0, ϕ0⟩, we recover (6.4). Step 5: On the operator derivative ∆̇0 b in the CR setting. In pseudo-Hermitian geometry one typically writes ∆λ b = −divλµ(∇λ b ) in terms of the λ-dependent horizontal gradient and divergence, the former depending on (θλ, Jλ) and the Levi form, the latter on the fixed measure µ (or, if the geometric volume µλ is preferred, one transports to the fixed Hilbert space via the unitary Uλf := ( dµλ dµ )1/2 f ; this falls under Kato’s unitary equivalence). Differentiating at λ = 0 gives a symmetric first-order differential operator ∆̇0 b whose coefficients are affine in the variations θ̇, J̇ and in the variation of the Levi form/Webster metric; the formula (6.4) then evaluates λ̇1(0) as the expectation of this operator in the ground state ϕ0. Remark 4 (On domains and regularity). Assumption (ii) is satisfied in two standard setups: (a) operator sense: there exists a common core D ⊂ W 2,2 H (M) such that ∆λ b |D depends C1 on λ in the graph norm; or (b) form sense: each ∆λ b is associated with a closed coercive form aλ on the common form domain V =W 1,2 H (M) and λ 7→ aλ is C1. In case (b) the above proof via forms applies verbatim and is often technically simpler in CR geometry where coefficients appear in divergence form. See [17, Ch. VI–VII]. Remark 5 (Role of torsion). The formula (6.4) is purely spectral/variational and holds regardless of torsion; torsion enters through the explicit expression of ∆̇0 b in terms of the Tanaka–Webster connection. In Sasakian (torsion-free) deformations, the same identity holds with a simpler ∆̇0 b (no first-order torsion terms). In general pseudo-Hermitian deformations, the torsion variation contributes linear terms to ∆̇0 b but the Hellmann– Feynman identity remains unchanged. 7. Conclusion In this work, we established Cheeger and Buser–type inequalities for the first positive eigenvalue of the sub-Laplacian on compact strictly pseudoconvex pseudo-Hermitian CR manifolds. These inequalities connect the spectral gap to a geometric invariant, the CR Cheeger constant, thereby extending classical Riemannian isoperimetric theory to the CR framework. Our approach combines horizontal BV methods, the co-area formula, and Poincaré inequalities within a unified sub-Riemannian analytic setting. This provides new insight into the interplay between geometry, torsion, and spectral properties in pseudo- Hermitian manifolds. The analysis also highlights how curvature and torsion quantitatively influence the constants appearing in isoperimetric and spectral inequalities. Future work A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 15 of 16 will focus on boundary analogues, higher eigenvalues, and the stability of the spectral gap under CR deformations. Acknowledgements The author expresses his sincere appreciation to the reviewers for their thorough evaluation of the manuscript and for their valuable comments and constructive suggestions, which have helped to improve the clarity and accuracy of this work. References [1] J. Cheeger. A lower bound for the smallest eigenvalue of the laplacian. In R. Gunning, editor, Problems in Analysis. Princeton Univ. Press, 1970. [2] P. Buser. A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4), 15(2):213–230, 1982. [3] G. B. Folland and E. M. Stein. Estimates for the ∂̄b-complex and analysis on the Heisenberg group. Comm. Pure Appl. Math., 27:429–522, 1974. [4] D. Jerison and J. M. Lee. The Yamabe problem on CR manifolds. J. Differential Geom., 25:167–197, 1987. [5] D. Jerison and J. M. Lee. Intrinsic CR normal coordinates and the CR Yamabe problem. J. Amer. Math. Soc., 1:1–41, 1988. [6] B. Franchi, R. Serapioni, and F. Serra Cassano. Rectifiability and perimeter in Carnot–Carathéodory spaces. Math. Ann., 321:479–531, 2001. [7] N. Garofalo and D.-M. Nhieu. Isoperimetric and Sobolev inequalities for Carnot– Carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math., 49:1081–1144, 1996. [8] F. Baudoin and N. Garofalo. Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc., 19:151– 219, 2017. [9] R. L. Frank and E. H. Lieb. Sharp constants in several inequalities on the Heisenberg group. Ann. of Math. (2), 176:349–381, 2012. [10] Luigi Ambrosio, Roberta Ghezzi, and Valentino Magnani. Bv functions and sets of finite perimeter in sub-riemannian manifolds. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 32(3):489–517, 2015. [11] D. Jerison and A. Sánchez-Calle. Subelliptic, second order differential operators. In Lecture Notes in Math., volume 1324, pages 46–77. Springer, 1987. [12] Sorin Dragomir and Giuseppe Tomassini. Differential Geometry and Analysis on CR Manifolds. 2006. [13] L. Saloff-Coste. Aspects of Sobolev-type inequalities. 289, 2002. [14] H. Rossi. Attaching analytic spaces to an analytic space along a pseudoconcave boundary. In Proc. Conf. Complex Analysis (Minneapolis, 1964), pages 242–256. Springer Lecture Notes, 1965. A. Ben Ahmed / Eur. J. Pure Appl. Math, 18 (4) (2025), 6985 16 of 16 [15] X. Huang and Y.-T. Siu. Non-embeddability of certain abstract CR manifolds. Invent. Math., 122:1–27, 1995. [16] H.Bosch, T.Gonzales, K.Spinelli, G.Udell, and Y.E. Zeytuncu. CR embeddability of quotients of the Rossi sphere via spectral theory. International Journal of Mathematics, 33(02):2250014, 2022. [17] T. Kato. Perturbation theory for linear operators. 1995.