An Optimal Gap Theorem in a Complete Strictly Pseudoconvex CR $(2n+1)$-Manifold

Abstract

In this paper, by applying a linear trace Li–Yau–Hamilton inequality for a positive (1, 1)-form solution of the CR Hodge–Laplace heat equation and monotonicity of the heat equation deformation, we obtain an optimal gap theorem for a complete strictly pseudoconvex CR $(2n+1)$-manifold with nonnegative pseudohermitian bisectional curvature and vanishing torsion. We prove that if the average of the Tanaka–Webster scalar curvature over a ball of radius r centered at some point o decays as $o\left( r^{-2}\right)$, then the manifold is flat.

Appendix

In this Appendix, we construct “nice” domains to avoid the possibility of the bad regularity for heat solutions in the case of degenerated parabolic systems. In fact, we will give a proof on existence and regularity result for $\left( 1,1\right)$-form $\phi$ of the Lichnerowicz-sub-Laplacian heat equation. In the proof of main theorem, one required some regularity of the heat solution in order to prove the mixed terms $\left( \bar{\partial } _{b}^{*}\Lambda \bar{\partial }_{b}+conj\right) \phi$ of (5.13) vanishing (then the monotonicity follows). While we construct heat solution on complete manifolds with exhaustion domains, we need the interior regularity at least $C^{2,\alpha }\left( \Omega _{\mu }\right)$ and boundary regularity as continuous function in $C\left( \bar{\Omega }_{\mu }\right)$. This requirement are needed for Arzela Ascoli theorem and integration by part in (5.15). In semigroup method, better regularity of evolution equation comes from the regularity of infinitesimal generator.

We denote $C^{2,\alpha }\left( \Omega ,\Lambda ^{1,1}\right)$ as $C^{2,\alpha }$ sections of $\Lambda ^{1,1}$ on bounded domain $\Omega$. In our case, it is $\Delta _{H}$ on Banach space $C^{2,\alpha }\left( \Omega ,\Lambda ^{1,1}\right) \cap C\left( \bar{\Omega },\Lambda ^{1,1}\right)$. Note $\Delta _{H}=-\frac{1}{2}\left( \square _{b}+\bar{\square }_{b}\right)$ . Here we denote u as solution of following Dirichlet problem

$$\begin{aligned} \square _{b}\phi =g \end{aligned}$$

(6.1)

for $g\in C^{\infty }\left( \Omega ,\Lambda ^{1,1}\right)$. First we state some results:

Theorem 6.1

(Kohn) Let M be a strictly pseudoconvex CR $(2n+1)$-manifold. If $1\le q\le n-1$ , then $\left\| \phi \right\| _{\frac{1}{2}}^{2}\le C\left[ \left( \square _{b}\phi ,\phi \right) +\left\| \phi \right\| _{0}^{2}\right]$ for $\phi \in C^{\infty }\left( \Lambda ^{0,q}\right) . \left\| .\right\| _{s}$ stands for the $L^{2}$ Sobolev norm of order s.

Remark 6.1

1. From the hypothesis in above theorem it requires $n\ge 2$. When $n=1$, one refers to [16].

2. Even though the operator $\square _{b}$ is not $\Delta _{H}$, in [17] (see p. 146) they actually prove the case for $\alpha =0$. Moreover, we have $\Delta _{H}=\mathcal {L}_{\alpha }$ with $\alpha =0$ up to lower order terms. Here $\mathcal {L}_{\alpha }=-\Delta _{b}+i\alpha T$.

The following is the interior and boundary regularity result by Jerison [16].

Theorem 6.2

Let U be the open subset of M containing no characteristic points of $\partial \Omega$. If $\psi ,\varphi \in C_{0}^{\infty }\left( U\right) , \psi =1$ in the neighborhood of the support of $\varphi$, and u satisfies (6.1) with $\psi g\in \Gamma _{\beta }\left( \bar{\Omega } ,\Lambda ^{0,q}\right)$, then $\varphi \phi \in \Gamma _{\beta +2}\left( \bar{\Omega },\Lambda ^{0,q}\right)$ and

$$\begin{aligned} \left\| \varphi \phi \right\| _{\Gamma _{\beta +2}}\le c\left( \left\| \psi g\right\| _{\Gamma _{\beta }}+\left\| \psi \phi \right\| _{L^{2}}\right) . \end{aligned}$$

When an isolated characteristic boundary point occurs, Jerison proved the regularity result when the neighborhood have strictly convexity property. The convexity is defined by Folland-Stein local coordinates $\Theta \left( p,-\right) :U\rightarrow \mathrm {H}^{n}$, and the boundary near point p is corresponding to graph $\tilde{t}=\sum \alpha _{i}\tilde{x}_{i}^{2}+\beta _{j}\tilde{y}_{j}^{2}+e\left( \tilde{x},\tilde{y}\right)$, where $e\left( \tilde{x},\tilde{y}\right) =O\left( \left| \tilde{x}\right| ^{3}+\left| \tilde{y}\right| ^{3}\right)$. Strictly convex means $\alpha _{i},\beta _{j}>0$ (see Eq. (7.4) and A.3 in [17]). In the following we state the theorem in the form we want. Reader who is confused can refer to Theorem 7.6, Proposition 7.11, and Corollary 10.2 in [17].T

Theorem 6.3

Let p be an isolated characteristic point on $\partial \Omega$ and in some neighborhood $U_{p}$ of p the geometry $U_{p}\cap \Omega$ is like the domain $\left\{ \left( x,y,t\right) :M_{c}\left( \left| x\right| ^{2}+\left| y\right| ^{2}\right) <t\right\}$ in the Heisenberg group, where $M_{c}$ a positive number . Then $\varphi \phi \in \Gamma _{\beta +2}\left( \bar{\Omega },\Lambda ^{0,q}\right)$, where the best $\beta$ depends on $M_{c}$. Moreover, as $M_{c}\nearrow \infty$, one can choose $\beta \nearrow \infty$.

Remark 6.2

In Theorem 6.3, one required $g\in \Gamma _{\beta }\left( \bar{\Omega } ,\Lambda ^{0,q}\right)$ for $\beta >2$. Moreover, $\beta$ has upper bound $\beta _{0}-2$ , where $\beta _{0}$ is an index related to the geometry of the boundary. In [17], they proved $M_{c}\nearrow \infty$, then $\beta _{0}\nearrow \infty$.

In order to construct a $C^{2,\alpha }$ Lichnerowicz-sub-Laplacian heat solution, we need the exhaustion domain which satisfy the property above. In the following we prove that it is possible by perturbing the boundary of exhaustion domain.

Theorem 6.4

For any given positive number $M_{c}$, there exists exhaustion domains $\Omega _{\mu }$ such that $\partial \Omega _{\mu }$ consist only isolated characteristic points with property as in Theorem 6.3 with given $M_{c}$.

Proof

We construct the exhaustion domain with smooth boundary arbitrarily. Since $\partial \Omega _{\mu }$ is compact, we define $\Xi _{\mu }$ the set consisting all the characteristic points. Then the closure of $\Xi _{\mu }$ is compact. At each point there exist coordinate $V_{p}$ such that we can express the boundary as $r\left( z,t\right) =t-q\left( z\right) +e\left( x,y\right)$ in $B_{p}\left( \varepsilon _{p}\right)$ for some $\varepsilon _{p}$ depend on p, where $q\left( z\right) =\alpha _{i}x_{i}^{2}+\beta _{j}y_{j}^{2}$ for some real numbers $\alpha _{i},\beta _{j}$. Since injective radius (with respect to some adapted metric) is uniformly bounded below on $\partial \Omega _{\mu }, \varepsilon _{p}$ can be chosen to not depend on p but $\mu$ only. These Folland-Stein coordinate neighborhoods form an open covering for $\bar{\Xi }_{\mu }$.

Now we claim there is a small modification to boundary so that $\bar{\Xi } _{\mu }$ contains only isolated characteristic points.

Assume $B_{p_{i}}\left( \varepsilon \right)$ are the covering of $\bar{\Xi } _{\mu }$, we can choose $\varepsilon _{1}<\varepsilon _{2}<\varepsilon$ such that $B_{p_{i}}\left( \varepsilon _{1}\right)$ are still a covering of $\bar{\Xi }_{\mu }$. We start at point $p_{1}$. First we deform the graph in the coordinate of $B_{p_{1}}\left( \varepsilon _{1}\right)$ to plane $t=0$ and smoothly attached to graph on $\partial B_{p_{i}}\left( \varepsilon _{2}\right)$. Under the deformation we keep point $p_{1}$ as the only characteristic point. This is possible by noticing that we only need to take $q\left( z\right)$ into consideration (because this term dominate all the other inside small ball. ) and we only need to consider the case in the Heisenberg group with graph $t=q\left( z\right)$ in $B_{p_{i}}\left( \varepsilon \right)$. We modify $q\left( z\right)$ into new one $\tilde{q} \left( z\right)$ by define $\tilde{q}\left( z\right) =-\max \limits _{\left| z\right| =\varepsilon _{2}}q\left( z\right)$ in $B_{p_{i}}\left( \varepsilon _{1}\right)$ and $\varphi \left( \left| z\right| ,\theta \right)$ in $B_{p_{i}}\left( \varepsilon _{2}\right) \backslash B_{p_{i}}\left( \varepsilon _{1}\right)$ where $\varphi \left( \left| z\right| ,\theta \right)$ is a smooth monotone function in $\left| z\right|$ for each $\theta$ such that the function smoothly attached to the value $q\left( z\right)$ on $\partial B_{p_{i}}\left( \varepsilon _{2}\right)$ and $\tilde{q}\left( z\right) =q\left( z\right)$ on $B_{p_{i}}\left( \varepsilon \right) \cap B_{p_{i}}^{c}\left( \varepsilon _{2}\right)$. This modification clearly imply the origin is the only characteristic point in $B_{p_{i}}\left( \varepsilon _{1}\right)$. Moreover, we can choose $\varphi \left( \left| z\right| ,\theta \right)$ very steep so that all the point $(z,q\left( z\right) )$ for $z\in B_{p_{1}}\left( \varepsilon _{2}\right) \backslash \left( 0,0\right)$ are noncharacteristic. We define the new domain as $\Omega _{\mu ,1}$. Specifically,

$$\begin{aligned} \Omega _{\mu ,1}=\left\{ \Omega _{\mu }\backslash B_{p_{1}}\left( \varepsilon _{2}\right) \right\} \cup \left( M\cap \left\{ \left( z,t\right) :t>\tilde{q}\left( z\right) -R\left( z,t\right) \text { for }z\in B_{p_{1}}\left( \varepsilon _{2}\right) \right\} \right) . \end{aligned}$$

Then we continue the same process on $p_{2}$, and the new domain is $\Omega _{\mu ,2}$. Observe that the process do not create new characteristic points but eliminate all the characteristic point inside $B_{p_{i}}\left( \varepsilon _{2}\right)$ except $p_{i}$. Continuing this process we are able to deform domain $\Omega _{\mu }$ into new one that only consist isolated characteristic points on the boundary with $M_{c}=0$.

To modify $M_{c}$ into any value we want is easier. One can do the same process by deforming the graph into parabolic. $\square$

For convenience, we call the domain in the above theorem a sweetsop domain.

Remark 6.3

The above theorem can be simplified if we can construct strictly convex domain in M. But the existence to this kind of exhaustion domain isn’t known yet.

We recall theorems from semigroup method. For the definition of analytic semigroup, one can refer to definition 12.30 in [39] ([38]). We cited the characterization of infinitesimal generator of analytic semigroups. Notation here X is Banach space and A is operator defined on X. Note A can be unbounded operator $A:D\left( A\right) \rightarrow X$, where $D\left( A\right)$ is a subset in X such that Ax can be defined. As before, we denote $\Gamma _{\beta }$ the Lipschitz classes associated with nonisotropic distance (referred [17]) and $\Gamma _{\beta }\left( \bar{ \Omega },\Lambda ^{1,1}\right)$ the restriction to $\bar{\Omega }$ of sections of $\Lambda ^{1,1}$ with coefficients in $\Gamma _{\beta }\left( \bar{\Omega }\right)$. We denote $\left\| {}\right\| _{\Gamma _{\beta }}$ the norm of Banach space $\Gamma _{\beta }\left( \bar{\Omega } ,\Lambda ^{1,1}\right)$, and $R_{\lambda }\left( A\right)$ as the inverse operator of $A_{\lambda }:=A-\lambda I$ as $A_{\lambda }$ is one-to-one. The resolvent set of the operator A is the subset of ${\mathbb {C}}$ that $R_{\lambda }\left( A\right)$ exists, bounded, and the domain is dense in X. When we apply, we let $X=\Gamma _{\beta }\left( \bar{\Omega } ,\Lambda ^{1,1}\right)$ and $A=\Delta _{H}$. Here we state general theorems for following evolution systems

$$\begin{aligned} \dot{u}=Au+f \end{aligned}$$

where $f\in X$.

Theorem 6.5

([39, Theorem 12.31]) A closed, densely defined operator A in X is the generator of an analytic semigroup if and only if there exists $\omega$ a real number such that the half-plane $\text {Re}\lambda >\omega$ is contained in the resolvent set of A and, moreover, there is a constant C such that

$$\begin{aligned} \left\| R_{\lambda }\left( A\right) \right\| \le \frac{C}{\left| \lambda -\omega \right| } \end{aligned}$$

(6.2)

for $\text {Re}\lambda >\omega$ and $\left\| .\right\|$ is the norm of X.

Theorem 6.6

([39, Theorem 12.33]) Let A be the infinitesimal generator of an analytic semigroup and assume that the spectrum of A is entirely to the left of the line $\text {Re}\lambda =\omega$. Then there exists a constant M such that

$$\begin{aligned} \left\| e^{At}\right\| \le Me^{\omega t}, \end{aligned}$$

where $\left\| .\right\|$ is the norm of X.

One can refer to Sect. 7.1 in [38] or page 421 in [10] for the application of semigroup theory for A is a strong elliptic operator. In our case, the missing boundary regularity is replaced by Theorem 6.3 (refer to [12, 13]). Then one can follow Stewart [40] and consider Hölder spaces as interpolation space [18] to obtain the resolvent estimates 6.2. As a result, the regularity of the parabolic systems follows by Theorem 6.6.

In conclusion, we are able to choose exhaustion domain with $\beta _{0}$ large enough, then follow theorem above, we can choose $\beta$ large enough to make sure the function space X is contained in $C^{2,\alpha }$. This is possible by relation $C^{\beta }\subset \Gamma _{\beta }\subset C^{\beta /2}$ as in 20.5, 20.6 of [11]. This completes the proof of Proposition 3.1.