A Lower Bound on the Spectrum of the Sublaplacian

Abstract

We establish a new lower bound on the first nonzero eigenvalue $\lambda _1 (\theta )$ of the sublaplacian $\Delta _b$ on a compact strictly pseudoconvex CR manifold $M$ carrying a contact form $\theta$ whose Tanaka–Webster connection has pseudohermitian Ricci curvature bounded from below.

Appendix: The Chang–Chiu Inequality

The purpose of Appendix is to give a proof of

$$\begin{aligned} 4 n \, \left\| u_0 \right\| _{L^2}^2 \le \frac{1}{n} \, \left\| \Delta _b u \right\| _{L^2}^2 + 4 \, \tau _0 \, \left\| \nabla ^H u \right\| _{L^2}^2 \end{aligned}$$

(91)

for any $u \in C^\infty (M, {\mathbb R})$ (compare² to (3.5) in [11], p. 270). This is referred to as the Chang–Chiu inequality. To prove (91) let us contract (13) by $u^\beta$ so that to obtain $u^\beta \nabla _0 u_\beta = u^\beta \nabla _\beta u_0 - A_{\alpha \beta } u^\alpha u^\beta$ or

$$\begin{aligned} u^\beta \, \nabla _0 u_\beta = \nabla _\beta \left( u_0 u^\beta \right) - u_0 \, \nabla _\beta u^\beta - A_{\alpha \beta } u^\alpha u^\beta . \end{aligned}$$

(92)

On the other hand (by (12)), $\nabla _\beta u^\beta = \nabla _{\overline{\beta }} u^{\overline{\beta }} - 2 i n \, u_0$ so that (by substitution into (92))

$$\begin{aligned} u^\beta \nabla _0 u_\beta + u_0 \, \nabla _{\overline{\beta }} u^{\overline{\beta }} = 2 i n \, u_0^2 - A_{\alpha \beta } u^\alpha u^\beta + \nabla _\beta \left( u_0 u^\beta \right) . \end{aligned}$$

(93)

Next (again by (13)), $u_0 \, \nabla _{\overline{\beta }} u^{\overline{\beta }} = \nabla _{\overline{\beta }} \left( u_0 u^{\overline{\beta }} \right) - u^{\overline{\beta }} \left( \nabla _0 u_{\overline{\beta }} + u_\gamma A^\gamma _{\overline{\beta }} \right)$ hence (by substitution of $u_0 \, \nabla _{\overline{\beta }} u^{\overline{\beta }}$ into (93))

$$\begin{aligned}&i \left( u^{\overline{\beta }} \, \nabla _0 u_{\overline{\beta }} - u^\beta \, \nabla _0 u_\beta \right) \nonumber \\&\quad = 2 n u_0^2 + i \left( A_{\alpha \beta } u^\alpha u^\beta - A_{\overline{\alpha }\overline{\beta }} u^{\overline{\alpha }} u^{\overline{\beta }} \right) + i \left\{ \nabla _{\overline{\alpha }} \left( u_0 u^{\overline{\alpha }} \right) - \nabla _\alpha \left( u_0 u^\alpha \right) \right\} \end{aligned}$$

(94)

(compare to (2.4) in Lemma 2.2, [11], p. 268). Calculations are performed with respect to an arbitrary local frame $\{ T_\alpha : 1 \le \alpha \le n \}$ in $T_{1,0}(M)$ (rather than a $G_\theta$-orthonormal frame, as in [11]). The next step is to evaluate the left-hand side of (94) in terms of the operator $P + \overline{P}$. One has $u_0 = (i/2n) \, \left( \nabla _\beta u^\beta - \nabla _{\overline{\beta }} u^{\overline{\beta }} \right)$ hence (by (13))

$$\begin{aligned} u^{\overline{\alpha }} \nabla _0 u_{\overline{\alpha }} = \frac{i}{2n} \, u^{\overline{\alpha }} \left( {{u_{\overline{\alpha }}}^{\overline{\gamma }}}_{\overline{\gamma }} - {{u_{\overline{\alpha }}}^\gamma }_\gamma \right) - A_{\overline{\alpha }\overline{\beta }} u^{\overline{\alpha }} u^{\overline{\beta }} . \end{aligned}$$

(95)

Using $P_{\overline{\alpha }} u \equiv {{u_{\overline{\alpha }}}^\gamma }_\gamma - 2 n i \, A_{\overline{\alpha }\overline{\beta }} u^{\overline{\beta }}$ the identity (95) becomes

$$\begin{aligned} i \, u^{\overline{\alpha }} \, \nabla _0 u_{\overline{\alpha }} = \frac{1}{2n} \, u^{\overline{\alpha }} \left( P_{\overline{\alpha }} u - {{u_{\overline{\alpha }}}^{\overline{\gamma }}}_{\overline{\gamma }} \right) . \end{aligned}$$

(96)

Let us take the complex conjugate of (96) and add the resulting equation to (96). We obtain

$$\begin{aligned} 2n i \, \left( u^{\overline{\alpha }} \, \nabla _0 u_{\overline{\alpha }} - u^\beta \, \nabla _0 u_\beta \right) = u^{\overline{\alpha }} P_{\overline{\alpha }} u + u^\alpha P_\alpha u - \left\{ u^{\overline{\alpha }} \, {{u_{\overline{\alpha }}}^{\overline{\gamma }}}_{\overline{\gamma }} + u^\alpha \, {{u_\alpha }^\gamma }_\gamma \right\} \end{aligned}$$

(97)

where $P_\alpha u \equiv {{u_\alpha }^{\overline{\gamma }}}_{\overline{\gamma }} + 2 n i A_{\alpha \beta } u^\beta$. Let us replace $u^\alpha \; {{u_\alpha }^\beta }_\beta + u^{\overline{\alpha }} \; {{u_{\overline{\alpha }}}^{\overline{\beta }}}_{\overline{\beta }}$ from (18) into (97). We obtain

$$\begin{aligned} 2ni \left( u^{\overline{\alpha }} \, \nabla _0 u_{\overline{\alpha }} - u^\alpha \, \nabla _0 u_\alpha \right)&= 2 \left( u^\alpha \, P_\alpha u + u^{\overline{\alpha }} P_{\overline{\alpha }} u \right) \nonumber \\&\quad - 2 n i \left( A_{\alpha \beta } u^\alpha u^\beta - A_{\overline{\alpha }\overline{\beta }} u^{\overline{\alpha }} u^{\overline{\beta }} \right) \nonumber \\&\quad + \left( \nabla ^H u \right) ( \Delta _b u). \end{aligned}$$

(98)

Finally substitution from (98) into (94) leads to

$$\begin{aligned}&2 \left( u^\alpha \, P_\alpha \!+\! u^{\overline{\alpha }} \, P_{\overline{\alpha }} u \right) \!+\! \left( \nabla ^H u \right) \left( \Delta _b u \right) \nonumber \\&\quad = 4 n^2 u_0^2 \!+\! 4 n i \left( A_{\alpha \beta } u^\alpha u^\beta \!-\! A_{\overline{\alpha }\overline{\beta }} u^{\overline{\alpha }\overline{\beta }} \right) \!+\! 2 n i \left\{ \nabla _{\overline{\alpha }} \left( u_0 u^{\overline{\alpha }} \right) \!-\! \nabla _\alpha \left( u_0 u^\alpha \right) \right\} .\qquad \quad \end{aligned}$$

(99)

Let us observe that

$$\begin{aligned}&i \left( A_{\alpha \beta } u^\alpha u^\beta - A_{\overline{\alpha }\overline{\beta }} u^{\overline{\alpha }\overline{\beta }} \right) = A\left( \nabla ^H u , \, J \nabla ^H u \right) ,\\&i \left\{ \nabla _\alpha \left( u_0 u^\alpha \right) - \nabla _{\overline{\alpha }} \left( u_0 u^{\overline{\alpha }} \right) \right\} = \mathrm{div} \left( u_0 \, J \nabla ^H u \right) , \end{aligned}$$

and $u^\alpha \, P_\alpha + u^{\overline{\alpha }} \, P_{\overline{\alpha }} u = g_\theta ^*(L u , \, d_b u )$ where $L = P + \overline{P}$. Then (99) becomes

$$\begin{aligned} 2 \, g_\theta ^*\left( L u , \, d_b u \right) + \left( \nabla ^H u \right) (\Delta _b u )&= 4 n^2 \, u_0^2 \nonumber \\&\quad + 4 n \, A \left( \nabla ^H u , \, J \nabla ^H u \right) - 2 n \, \mathrm{div} \left( u_0 \, J \nabla ^H u \right) .\nonumber \\ \end{aligned}$$

(100)

Let us integrate over $M$ and use Green’s lemma. Then (by Lemma 1)

$$\begin{aligned}&- 2 \int \limits _M (P_0 u) u \, \Psi _\theta + \int \limits _M \left( \nabla ^H u \right) (\Delta _b u) \, \Psi _\theta \nonumber \\&\quad = 4 n^2 \, \left\| u_0 \right\| _{L^2}^2 + 4 n \int \limits _M A \left( \nabla ^H u , \, J \nabla ^H u \right) \, \Psi _\theta . \end{aligned}$$

(101)

Also (again by Green’s lemma)

$$\begin{aligned} \int \limits _M \left( \nabla ^H u \right) (\Delta _b u ) \; \Psi _\theta = \int \limits _M \left( \Delta _b u \right) ^2 \; \Psi _\theta = \left\| \Delta _b u \right\| _{L^2}^2 . \end{aligned}$$

Finally as $P_0$ is nonnegative (87) and (101) lead to (91). Q.e.d.