Pseudo-harmonic Maps from Complete Noncompact Pseudo-Hermitian Manifolds to Regular Balls

Abstract

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls of Riemannian manifolds.

Appendix

This section will deduce Theorems 2.4 and 5.1 by the theory of subelliptic analysis. Suppose that $(M, \theta )$ is a pseudo-Hermitian manifold of real dimension $2m +1$. Let $\Omega$ be a coordinate neighborhood in M and $\{ e_B \}_{B=1}^{2m}$ be an orthonormal basis of $HM \big |_\Omega$ with $J e_i = e_{i +m}$ for $i = 1, 2, \dots , m$. Since

$$\begin{aligned} -\, \theta ([e_i, J e_i]) = \mathrm{d} \theta (e_i, J e_i) = G_\theta (e_i, e_i) = 1, \quad \text{ for } i = 1, 2, \dots , m, \end{aligned}$$

then each $[e_i, J e_i]$ is transversal with horizontal distribution which implies that HM satisfies the strong bracket generating hypothesis. Moreover, by identifying $\Omega$ with a domain in ${\mathbb {R}}^{2m+1}$, the vector fields $\{e_1, \dots , e_{2m} \}$ satisfy the Hörmander’s condition. Let $e_B^*$ be the formal adjoint of $e_B$. For any $u \in C^\infty (\Omega )$, we have

$$\begin{aligned} \Delta _b u = - \sum _{B =1}^{2m} e_B^* e_B u, \end{aligned}$$

which shows that the sub-Laplacian operator is subelliptic. One can refer to [12, Sect. 2.2] for more discussions. Since Tanaka–Webster connection preserves the horizontal distribution, then the higher-order horizontal covariant derivative on $\Omega$ can be expressed as follows:

$$\begin{aligned} \nabla _b^l u (e_{B_1}, \cdots , e_{B_l})&= \nabla _{e_{B_l}} \bigg [ \nabla ^{l-1} u (e_{B_1}, \cdots , e_{B_{l-1}}) \bigg ] \\&\quad - \sum _{i = 1}^{l} \nabla ^{l-1} u (e_{B_1}, \cdots , \nabla _{e_{B_l}} e_{B_i}, \cdots , e_{B_l}) \\&= \cdots \\&= e_{B_l} e_{B_{l-1}} \cdots e_{B_1} u + \text{ lower } \text{ order } \text{ terms }, \end{aligned}$$

for any $B_1, \cdots , B_l \in \{1, 2, \cdots , 2m\}$, which implies that the $S^p_k$-norm on $\Omega$ is equivalent with the local Folland–Stein Sobolev norm (cf. [12, p. 193]). Hence local results of subelliptic analysis always hold for the sub-Laplacian operator on a coordinate neighborhood of pseudo-Hermitian manifolds. By partition of unity, the domain can be generalized to a relatively compact domain in a pseudo-Hermitian manifold. Let us use this idea to prove Theorem 2.4 by the following local version.

Theorem 6.1

([12, Theorem 3.17] and [23, Theorem 16]) Suppose that $(M,\theta )$ is a pseudo-Hermitian manifold and $\Omega \Subset M$ is a coordinate neighborhood. Assume that $u, v \in L_{loc}^1 (\Omega )$ and $\Delta _b u = v$ in the distribution sense. For any $\chi \in C^\infty _0 (\Omega )$, if $v \in S^p_k (\Omega )$ with $p >1$ and $k \in {\mathbb {N}}$, then $\chi u \in S^p_{k+2} (\Omega )$ and

$$\begin{aligned} ||\chi u||_{S^p_{k+2} (\Omega )} \le C_{\chi } \left( ||u||_{L^p (\Omega )} + ||v||_{S^p_k (\Omega )} \right) , \end{aligned}$$

(6.1)

where $C_{\chi }$ only depends on $\chi$.

Proof of Theorem 2.4

Let $\{\Omega _\alpha \}$ be a finite open cover of ${\text {supp}} \chi$ and $\{\chi _\alpha \}$ be a partition of unity subordinating to $\{\Omega _\alpha \}$. Since $\Delta _b u = v$ holds in each $\Omega _\alpha$, Theorem 1 guarantees that

$$\begin{aligned} || \chi _\alpha \chi u||_{S^p_{k+2} (\Omega _\alpha )} \le C_{\chi _\alpha \chi } \left( ||u||_{L^p (\Omega _\alpha )} + ||v||_{S^p_k (\Omega _\alpha )} \right) , \end{aligned}$$

which implies that

$$\begin{aligned} ||\chi u||_{S^p_{k+2} (\Omega )} \le \sum _\alpha ||\chi u||_{S^p_{k+2} (\Omega _\alpha )} \le \left( \sum _\alpha C_{\chi _\alpha \chi } \right) \left( ||u||_{L^p (\Omega )} + ||v||_{S^p_k (\Omega )} \right) . \end{aligned}$$

The proof is finished by setting $C_\chi = \sum _\alpha C_{\chi _\alpha \chi }$. $\square$

Next let us prove Theorem 5.1.

Proof of Theorem 5.1

Under the exponential map at $p_0 \in N$, the regular ball $B_D = B_D (p_0)$ is diffeomorphic to the ball $B_D$ with radius D and centered at the origin in ${\mathbb {R}}^n$ where $n = {\text {dim}} N$. Let $\{z^i\}_{i=1}^n$ be the geodesic normal coordinates at $p_0$ and $f^i = z^i \circ f$ be the components of a function $f : M \rightarrow B_D$. Denote

$$\begin{aligned} {\mathcal {S}} = \left\{ f \in S^2_1 (M, {\mathbb {R}}^n) \bigg | \, f - \varphi \in S^2_{1,0} (M, {\mathbb {R}}^n), \ \sup _M |f| \le D \right\} , \end{aligned}$$

where $|\cdot |$ is the Euclidean norm in ${\mathbb {R}}^n$. Consider the minimizing problem

$$\begin{aligned} \lambda = \inf _{f \in {\mathcal {S}}} E_H (f) = \inf _{f \in {\mathcal {S}}} \int _M h_{ij} (f) \langle \nabla _b f^i, \nabla _b f^j, \rangle \end{aligned}$$

(6.2)

where $h_{ij} = h (\frac{\partial }{\partial z^i}, \frac{\partial }{\partial z^j})$. Since $\varphi \in {\mathcal {S}}$, then $\lambda$ is finite. Let $\{f_s\}_{s =1}^\infty$ be a minimizing sequence of (6.2) which have uniform $S^2_1$-norm bound. By CR compact embedding theorem of Folland–Stein space (cf. Theorem 3.15 in [12]), there are a $f \in S^2_1 (M , {\mathbb {R}}^n)$ and a subsequence of $\{f_s\}$ (also denoted by $\{f_s\}$) such that

• (i) $f_s \rightarrow f$ strongly in $L^2 (M, {\mathbb {R}}^n)$;

• (ii) $f_s \rightharpoonup f$ weakly in $S^2_1 (M, {\mathbb {R}}^n)$.

By (1), $f_s$ converges to f almost everywhere on M which implies that $|f| \le D$; by (1), $f - \varphi \in S^2_{1,0} (M, {\mathbb {R}}^n)$ which is closed in $S^2_1 (M, {\mathbb {R}}^n)$. Hence $f \in {\mathcal {S}}$.

We claim that

$$\begin{aligned} E_H (f) \le \liminf _{s \rightarrow \infty } E_H (f_s). \end{aligned}$$

(6.3)

It suffices to show that for any domain $\Omega \subset M$ with an orthonormal basis $\{e_A\}_{A = 1}^{2m}$ of $HM \big |_{\Omega }$,

$$\begin{aligned} \sum _{i, j, A} \int _\Omega h_{ij} (f) e_A f^i \, e_A f^j \le \liminf _{s \rightarrow \infty } \sum _{i, j, A} \int _\Omega h_{ij} (f_s) e_A f^i_s \, e_A f^j_s. \end{aligned}$$

(6.4)

For any $\varepsilon > 0$, since $f^i \in S^2_1 (\Omega )$ and $f_s^i \rightarrow f^i$ strongly in $L^2 (\Omega )$, there is a compact set $K \subset \Omega$ such that

$$\begin{aligned} \sum _{i, j, A} \int _{\Omega \setminus K} h_{ij} (f) e_A f^i \, e_A f^j < \varepsilon \quad \text{ and } \quad f^i_s \rightrightarrows f^i \quad \text{ on } K, \end{aligned}$$

where “$\rightrightarrows$” means “uniform convergence”. The positivity of $(h_{ij})$ implies that

$$\begin{aligned} 0&\le \sum _{i, j, A} h_{ij} (f_s) e_A (f^i_s - f^i) \, e_A (f^j_s - f^j) \\&= \sum _{i, j, A} h_{ij} (f_s) e_A f^i_s \, e_A f^j_s - \sum _{i, j, A} h_{ij} (f_s) e_A f^i \, e_A f^j \\&\quad - 2 \sum _{i, j, A} h_{ij} (f_s) e_A f^i \, e_A (f^j_s - f^j) , \end{aligned}$$

which yields that

$$\begin{aligned} \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i_s \, e_A f^j_s&\ge \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i \, e_A f^j \nonumber \\&\quad + \,2 \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i \, e_A (f^j_s - f^j) \nonumber \\&= \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i \, e_A f^j \nonumber \\&\quad + \,2 \sum _{i, j, A} \int _K (h_{ij} (f_s) - h_{ij} (f)) e_A f^i \, e_A (f^j_s - f^j) \nonumber \\&\quad +\, 2 \sum _{i, j, A} \int _K h_{ij} (f) e_A f^i \, e_A (f^j_s- f^j). \end{aligned}$$

(6.5)

For the first term of (6.5), since $f_s^i \rightrightarrows f^i$ on K, then by mean value theorem, we have

$$\begin{aligned}&\left| \sum _{i, j, A} \int _K (h_{ij} (f_s)- h_{ij} (f)) e_A f^i \, e_A f^j \right| \\&\le \sum _{i, j, k, A} \max _{B_D} \left| \frac{\partial h_{ij}}{\partial z^k} \right| \int _K |f_s^k - f^k| \, |e_A f^i| \, |e_A f^j| \rightarrow 0, \end{aligned}$$

as $s \rightarrow \infty$, which implies that

$$\begin{aligned} \lim _{s\rightarrow \infty } \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i \, e_A f^j = \sum _{i, j, A} \int _K h_{ij} (f) e_A f^i \, e_A f^j. \end{aligned}$$

(6.6)

Similarly, since $e_A f^j_s$ and $e_A f^j$ are uniformly bounded in $L^2 (K)$, then

$$\begin{aligned} \lim _{s \rightarrow \infty } \sum _{i, j, A} \int _K (h_{ij} (f_s) - h_{ij} (f)) e_A f^i \, e_A (f^j_s - f^j) = 0. \end{aligned}$$

(6.7)

For the third term of (6.5), define an operator $T_A : S^2_1 (M) \rightarrow L^2 (K)$ by

$$\begin{aligned} T_A (u) = e_A u \big |_K. \end{aligned}$$

$T_A$ is continuous due to the following calculation:

$$\begin{aligned} || T_A (u) ||_{L^2 (K)}^2 = \int _K | e_A u |^2 \le \int _M | \nabla _b u |^2 \le || u ||_{S^2_1(M)}^2. \end{aligned}$$

Since any continuous operator between two Banach spaces preserves weak convergence, then $e_A f^i_s \rightharpoonup e_A f^i$ weakly in $L^2 (K)$ for any A and i. Hence

$$\begin{aligned} \lim _{s \rightarrow \infty } \sum _{i, j, A} \int _K h_{ij} (f) e_A f^i \, e_A (f^j_s- f^j) = 0 . \end{aligned}$$

(6.8)

Using (6.6), (6.7) and (6.8), we find that

$$\begin{aligned} \sum _{i, j, A} \int _K h_{ij} (f) e_A f^i \, e_A f^j \le \liminf _{s \rightarrow \infty } \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i_s \, e_A f^j_s, \end{aligned}$$

which implies that

$$\begin{aligned} \sum _{i, j, A} \int _\Omega h_{ij} (f) e_A f^i \, e_A f^j&\le \sum _{i, j, A} \int _K h_{ij} (f) e_A f^i \, e_A f^j + \varepsilon \\&\le \liminf _{s \rightarrow \infty } \sum _{i, j, A} \int _K h_{ij} (f_s) e_A f^i_s \, e_A f^j_s + \varepsilon \\&\le \liminf _{s \rightarrow \infty } \sum _{i, j, A} \int _\Omega h_{ij} (f_s) e_A f^i_s \, e_A f^j_s + \varepsilon . \end{aligned}$$

By taking $\varepsilon \rightarrow 0$, we obtain (6.4) and thus $E_H (f) \le \lambda$.

Obviously, $E_H (f) \ge \lambda$ and then $E_H (f) = \lambda$ which shows that f has the minimal horizontal energy in ${\mathcal {S}}$ and satisfies

$$\begin{aligned} \Delta _b f^i + \Gamma ^i_{jk} (f) \langle \nabla _b f^j, \nabla _b f^k \rangle = 0 , \end{aligned}$$

in the distribution sense. By applying Theorem 2 in [15] to f on each coordinate neighborhood $\Omega \Subset M$, we obtain the interior continuity of f. $\square$