Abstract
Let \(B_n\) be the unit ball in 
with the Bergman metric g and h is the Bergman metric on \(B_m\). Let \(u: (B_n, g)\rightarrow (B_m, h)\) be any harmonic map with \(\phi _0=u|_{\partial B_n}\in C^\infty (\partial B_n, \partial B_m)\). In this paper, we provide an asymptotic expansion formula for the above harmonic map u for a large class of \(\phi _0\in C^\infty (\partial B_n, \partial B_m)\).
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1 Introduction
The theory of the existence, regularity, and the rigidity for the proper harmonic maps between balls in hyperbolic metrics has been studied by Li and Tam [11,12,13] and Li and Wang [14, 15], Wang [28], etc. The complex case in Bergman metrics was studied by Donanelly [3] and Li and Ni [18] and Li and Simon [19], etc.. The main purpose of the paper is to study the asymptotic expansions for proper harmonic maps between balls in Bergman metrics in \({\mathbb {C}}^n\) and \({\mathbb {C}}^m\), respectively.
Let \(B_{n}\) be the unit ball in \({\mathbb {C}}^{n}\). The Bergman kernel on \( B_{n}\) is given by \(K(z,w)=(1-\langle z,w\rangle )^{-n-1}\). The Bergman metric is defined by
The Laplace–Beltrami operator \(\Delta _g\) in the Bergman metric in \(B_{n}\) is defined by
Let h be the Bergman metric on the unit ball \(B_{m}\) in \({\mathbb {C}}^{m}\) with the Christoffel symbols \(\Gamma _{t\gamma }^{s}\).
A \(C^{2}\) map \(u:B_{n}\rightarrow B_{m}\) is harmonic in Bergman metrics if the tension field
If \(u:B_{n}\rightarrow B_{m}\) is a proper harmonic map and if \(\phi _{0}(z)=u(z)|_{ \partial B_n} \in C^1(\partial B_{n})\), then
where
Let
and
Let \(\phi _{0}:\partial B_{n}\rightarrow \partial B_{m}\). The Dirichlet boundary value problem for a proper harmonic map in Bergman metric is given by
Assuming that \(\phi _0 \in C^\infty (\partial B_n)\) and \(E_b[\phi _0]\ne 0\) on \( \partial B_n\), the existence and regularity were studied by Li and Tam [11,12,13], Li and Wang [14, 15], Wang [28] for real case. The complex case was studied by Donnelly [3] (also see Li and Ni [18]). Their result can be stated as follows:
Theorem 1.1
Let \(\phi _0 :\partial B_{n}\rightarrow \partial B_{m}\) be a \( C^{\infty }\) map with \(E_{b}[\phi _0 ]\ne 0\) and satisfying (1.4). Then the Dirchlet boundary value problem (1.8) has a unique solution \(u\in C^{n,\alpha }({\overline{B}}_{n})\) for any \(0<\alpha <1\).
The question about the structure of u associated with its regularity is always interesting, the problem is the so-called the asymptotic expansion of u.
For the linear case: The asymptotic expansion for the solution of
was studied by Graham [5]. He proved the following theorem.
Theorem 1.2
If \(\phi \in C^{\infty }(\partial B_{n})\), then the unique solution u of (1.9) can be written by
where \(A,B\in C^{\infty }({\overline{B}}_{n})\).
For the fully nonlinear equation, the Fefferman equation with \(\rho >0\) is given by
where \(\Omega \) is a smoothly bounded pseudoconvex domain in \({\mathbb {C}} ^{n} \). The uniqueness and approximating solution with \(-\log (\rho )\) being plurisubharmonic were given by C. Fefferman [4] and the existence and uniqueness and regularity were established by Cheng and Yau [2]. When \( \Omega \) is strongly pseudoconvex, the maximum regularity is \(C^{n+1,\alpha }\) up to boundary for any \(0<\alpha <1\). The asymptotic expansion for the unique solution u was given by Lee and Melrose [9].
The harmonic map equation is a system of semi-linear equations. The question was asked by Li and Tam [12]: Does one have the asymptotic expansion for the solution of (1.8) similar to the one in [5] and [9]? The first purpose of this paper is to study this question. In order to state our result, we assume that \(\phi _0\in C^\infty (\partial B_n)\) and let
be a \((2m)\times (2n)\) matrix-valued function on \(\partial B_n\). For any \(f \in C(\partial B_n, {\mathbb {C}}^{2m})\), we let \(P_0[f](z)\) be the orthogonal projection of f(z) onto the range of \(h[\phi _0](z) h[\phi _0]^*(z)\). It is easy to see that if \(h[\phi _0] h[\phi _0]^* \) has constant rank, then \(P_0\) maps \( C^\infty (\partial B_n, {\mathbb {C}}^{2m})\) into \( C^\infty (\partial B_n, {\mathbb {C}}^{2m})\). Now we are ready to state our theorem.
Theorem 1.3
Let \(\phi _{0}:\partial B_{n}\rightarrow \partial B_{m}\) be a \( C^{\infty }\) map satisfying (1.4). If \(|\partial _{b}\phi _{0}|| {\overline{\partial }}_{b}\phi _{0}|>0\) on \(\partial B_{n}\) and \(P_0 (C^\infty (\partial B_n, {\textbf{C}}^{2m}))\subset C^\infty (\partial B_n, {\textbf{C}}^{2m})\), then the solution u of (1.8) has the following asymptotic expansion:
where \(\phi \) and \(\psi _{\ell }\in C^{\infty }({\overline{B}}_{n})\). Here, the infinite sum is in the sense of asymptotic expansion.
Note that if \(|\partial _b \phi _0 | |{\overline{\partial }}_b \phi _0|=0\) on \(\partial B_n\), the following theorem was proved by Li and Ni [18] on \(\partial B_n\) and by Li and Son [20] on more general torsion-free pseudo-Hermitian closed manifolds.
Theorem 1.4
If \(\phi \in C^2(\partial B_n)\) and if \(|\partial _b \phi _0| |{\overline{\partial }}_b \phi _0 |=0\) on \(\partial B_n\), then \(\phi _0\) is either CR or anti-CR on \(\partial B_n\).
The second purpose of the paper is looking for a sufficient and necessary condition on any boundary map \(\phi _0:\partial B_n\rightarrow \partial B_m\) so that it becomes either CR or anti-CR on \(\partial B_n\). The results are stated in Theorem 2.2. Since the computation of \(\tau ^s[u]\) is very complicated, in order to prove the asymptotic expansion formula, we divide our computations into several parts which are presented in Sections 3–5. The proof of Theorem 1.3 is given in Section 6. In Section 7, we will prove a technical proposition (Proposition 6.3). In Section 8, we provide a theorem, with assumption that \(h[\phi _0] h[\phi _0]^*\) has rank \(2m-2\), the harmonic map u has better regularity.
2 Preliminary
On the unit ball \(B_{n}\subset {\mathbb {C}}^{n}\), we define the radial first order differential operators:
the tangential first order differential operators:
For any \(C^{1}\) map \(\phi :{\overline{B}}_{n}\setminus \{0\}\rightarrow {\overline{B}}_{m}\setminus \{0\}\), we define
Lemma 2.1
Let \(\phi :\partial B_{n}\rightarrow \partial B_{m}\) be a \( C^{2}\) map with
Then
and
Proof
Notice that
one has
This implies (2.5) and (2.6) hold.
Since \(|\phi |^{2}=1\) on \(\partial B_{n}\) and \(\langle Y_{j}\phi ,\phi \rangle =0\), one has \(\left\langle {\overline{Y}}_{j}\phi ,\phi \right\rangle =0\). Therefore,
These imply
By (2.5) and (2.8) , one obtains that
and
Therefore, the proof of the lemma is complete.
\(\square \)
Theorem 2.2
Let \(\phi :\partial B_{n}\rightarrow \partial B_{m}\) be a \( C^{2}\) map satisfying (2.4). Then \(\phi \) is CR or anti-CR if and only if \(\phi \) satisfies
Proof
By Lemma 2.1, one has
and
Therefore,
implies
But the integrand is a nonnegative continuous function on \(\partial B_{n}\), one can easily see that
This implies that for any \(z\in \partial B_{n}\), one has either \(|Y\phi (z)|=0\) or \(|{\overline{Y}}\phi (z)|=0\). Therefore,
Applying Theorem 3.1 in [18] or Lemma 4.2 in [20], one has \( {\overline{\partial }}_{b}\phi =0\) on \(\partial B_{n}\) or \({\overline{\partial }} _{b}{\overline{\phi }}=0\) on \(\partial B_{n}\), which means either \(\phi \) is CR or \({\overline{\phi }}\) is CR.
On the other hand, if \(\phi \) is CR or \({\overline{\phi }}\) is CR, then \( {\overline{\partial }}_{b}\phi =0\) or \({\overline{\partial }}_{b}{\overline{\phi }} =0\) on \(\partial B_{n}.\) By (2.10) and (2.11) , one can easily show that (2.9) holds and the proof of the theorem is complete. \(\square \)
We state and prove the following proposition for future application.
Proposition 2.3
Let \(\phi :\partial B_{n}\rightarrow \partial B_{m}\) be a \( C^{2}\) map satisfying (2.4). Then
Proof
Since
the proof is complete. \(\square \)
3 Reformulation of the harmonic map equations
In this section, we introduce more notations and prove more lemmas which will be used later.
Definition 3.1
A differential operator \({\mathcal {D}}\) on \(C^{2}({\overline{B}}_{n} \setminus \{0\})\) is called a boundary operator if for every \(\phi \in C^{2}({\overline{B}} _{n} \setminus \{0\})\) with \(\phi (z)=\phi (\rho z)\), one has \({\mathcal {D}}\phi (z)=({\mathcal {D}} \phi )(\rho z) \).
Let
and
Proposition 3.2
With the above notations, one has
Proof
One can verify that \(R,{\overline{R}},R_{0}\),and \({\overline{R}}_{0}\) satisfy the Leibnitz’s rule. Moreover, by (2.7)
Since
and
one has
Therefore, the proof is complete. \(\square \)
Lemma 3.3
\(Y_{j},{{\mathcal {L}}}_{0}\), R, \({\overline{R}}\), \({{\mathcal {R}}} ^{2}\), and \(T^{2}\) are boundary operators.
Proof
Let \(\phi (z)=\phi (sz)\). Then
Similarly, one has \({\overline{R}}\phi (z)=({\overline{R}}\phi )(sz)\), \(R {\overline{R}}\phi (z)=(R{\overline{R}}\phi )(sz)\), T and \(T^{2}\) are boundary operators.
Since
one has
This proves that \({{\mathcal {L}}}_{0}\) is a boundary operator. Therefore, the lemma is proved. \(\square \)
Proposition 3.4
Let \(u\in C^{2} (B_{n},B_{m}) .\) Then u is harmonic if and only if
where
and
Proof
Let h be the Bergman metric on \(B_{m}\). Then the Christoffel symbols of \( (B_{m},h)\) are:
By (1.3), it is easy to see that u is harmonic in Bergman metrics if and only if
Let
Then u is harmonic if and only if (3.5) holds.
Note that
one has
Since
one has
Therefore, the proof of the proposition is complete. \(\square \)
4 Computation of \(( 1-\left| u\right| ^{2}) 4|z|^2Lu \)
4.1 Computation of the Linear Part: \(4|z|^2 L u\)
Let \(\phi _{k}\) be radial and smooth, and let
Define
and for \(k\ge 1\) with \(\phi _{-1}=0\),
Lemma 4.1
Let \(\phi \in C^{\infty }( {\overline{B}}_{n},{\overline{B}} _{m}) \) with the asymptotic expansion (4.1) near \(\partial B_{n}.\) Then
Proof
Notice that
and
we have
Since \({\mathcal {L}}_{0}\) and T are boundary operators and \({{\mathcal {R}}} \phi _{k}=0\), by (4.5) , ( 4.6) and (4.7) ,
Then
Therefore, the proof of the lemma is complete. \(\square \)
We will use the following notations:
and
Let
and
where \(D_{k}\) is defined by (4.2) and (4.3),
and
Notice that
and
Let \(v_{\ell }=\psi _{\ell }(\log (-r))^{\ell }.\) Then
Therefore,
Notice that \(\psi _{\ell ,k}=0\) when \(k<\kappa _{\ell }=\ell (n+1)\) and
where \(\xi _{\ell +1,k}\) is defined in (4.12) . Moreover,
where \(\zeta _{\ell +1,k}\) is defined in (4.14).
where \(\eta _{\ell +2,k}\) is defined in (4.13). Therefore, with \(\ell \ge 0\),
and
Therefore,
where \({\tilde{\xi }}_{\ell +1,k}\) is defined in (4.15) .
As a summary, we have proved the following proposition.
Proposition 4.2
Let \(\psi _{\ell ,k}=0\) when \(k<\kappa _{\ell }\) and let
in the sense of asymptotic expansion. Then
where \({\mathcal {D}}_{k}\) and \(\mathcal {{\mathcal {D}}}_{\ell ,k}\) are defined in (4.10) and (4.11).
4.2 Computation of \((1-|u|^2) 4|z|^2 L u\)
Define
and
Proposition 4.3
Let u be the map defined by (4.17) with \( \left| \phi _{0}\right| =1\), then
Proof
Since
by Proposition 4.2,
The proof of the proposition is complete. \(\square \)
It is easy to prove the following lemma:
Lemma 4.4
Let u be the map defined by (4.17). Let \({\mathcal {H}}\) and \({\mathcal {K}}_{{}}\) be operators on u such that
and
Then
where
and
5 Computation of P[u]
Since \({\mathcal {R}}\left( r^{k}\right) =kr^{k-1}(1+r)\),
and \(\psi _{\ell ,k}=0\) when \(k<\kappa _{\ell },\) one has
where
and
Since \(F_{\ell ,k}\left[ u\right] =0\) when \(k<\kappa _{\ell }-1\), applying Lemma 4.4, we have
Then
where
and
Let
and let
Then, by similar computations, one has that
and
Using equations (5.6), (5.13), (5.14), and (1.4), we have proved the following proposition.
Proposition 5.1
With notations above, one has
6 Proof of Theorem 1.3
Let
which is given by (4.17), where \(\psi _{\ell ,k}=0\) for \( k<\kappa _{\ell }=\ell (n+1)\). It is well known from [3] and [18] that if \( u\in C^2({\overline{B}}_n)\) is proper harmonic with \(u|_{\partial B_n}=\phi _0\), then
Combining Proposition 4.3 and Proposition 5.1, one has proved the following theorem
Theorem 6.1
Under the assumption (6.1), one has
where \(B_{k}\) and \(B_{\ell ,k}\) are given by (4.21) and (4.22); \(E_{k}\) and \(E_{\ell ,k}\) are given by (5.7) and (5.8); \( G_{k},G_{\ell ,k}\) are given by (5.9) and (5.10) and \(C_{k}\) and \(C_{\ell ,k}\) are given by (5.11), (5.12).
From the definition of \(B_{\ell ,k},E_{\ell ,k},G_{\ell ,k}\), and \(C_{\ell ,k} \), one has
Theorem 6.1 implies that if u is proper harmonic then (6.1) and
hold. We will solve \(\phi _k \) and \(\psi _{\ell , k}\) through the above system of equations.
6.1 \(B_{k}+E_{k}-G_{k}+C_{k}=0\) when \(1\le k\le n\)
Since
and
where \(e_{k}=k\left( n-k\right) .\)
For \(E_k\), one has
and
For \(G_{k}\), one has
and if \(k>1,\) then
For \(C_{k}\), one has
where
6.1.1 Case 1. \(k=1\)
Since \(B_{1}+E_{1}-G_{1}+C_{1}=0,\) one has
Since \(\langle Y_{j}\phi _{0},\phi _{0}\rangle =\langle {\overline{Y}}_{j}\phi _{0},\phi _{0}\rangle =0\) and \(D_{0}={{\mathcal {L}}}_{0}\phi _{0}+2(n-1)\phi _{1}\), one has
Notice that \(E_{b}[\phi _{0}]>0,\ {\mathcal {A}}_{1}=\lim _{r\rightarrow 0^-}\frac{ 1-\left| u\right| ^{2}}{(-r)}>0\) and \(\langle T\phi _{0},\phi _{0}\rangle ^{2}\ge 0\), one can easily see that \(A_{1,0}\) is real and \({ {\mathcal {A}}}_{1}=2A_{1,0}\). Therefore,
and thus
Therefore,
Let
Then
Therefore, (6.9) implies that
where
Notice that \(n{{\mathcal {A}}}_{1}I_{2m}+2h[\phi _{0}]h[\phi _{0}]^{*}\) is invertible, we have
6.1.2 Case 2. \(1<k\le n\)
Notice that
Then
implies that
In particular,
This implies
and
Therefore, with \(k(n+2-k)+n-1=e_{k}+2k+n-1\),
Therefore
This implies
and
6.2 Case: \(k\ge n+1\)
6.2.1 \(B_{\ell , k}+E_{\ell , k}-G_{\ell ,k}+C_{\ell , k}=0 \)
For \(\kappa _{\ell }\le k<\kappa _{\ell +1}\), notice that
and
where
When \(k<\kappa _{\ell +1}\), one has
where
Since
where
In particular,
and
Then
6.2.2 \(B_{1, \kappa }+E_{1, \kappa }-G_{1, \kappa }+C_{1, \kappa }=0\)
Let \(\kappa =\kappa _{1}\). Then from the previous subsection, one has
Notice that
Thus \(\langle X_{\kappa }, \phi _{0}\rangle =0\) if and only if
Therefore,
With the assumption \(\langle \psi _{1,\kappa },\phi _{0}\rangle =0\), one has
Then \(X_{\kappa }=0\) if and only if
Proposition 6.2
If \(h[\phi _{0}]h[\phi _{0}]^{*}\) has rank \(2m-2\) and if \(\langle \psi _{1,\kappa },\phi _{0}\rangle =0\) and \(h[\phi _{0}]h[\phi _{0}]^{*}[\psi _{1,\kappa }^{t},{\overline{\psi }}_{1,\kappa }^{t}]^{t}=0\) then \(\psi _{1,\kappa }=0\).
Proof
Since \(\langle \psi _{1,\kappa },\phi _{0}\rangle =0\), one has that \([\psi _{1,\kappa }^{t},{\overline{\psi }}_{1,\kappa }^{t}]\) is orthogonal to \([\phi _{0}^{t},0^{t}]\) and \([0^{t},{\overline{\phi }} _{0}^{t}]\) and \(h[\phi _{0}]h[\phi _{0}]^{*}[\phi _{0}^{t},0^{t}]^{t}=0\) and \(h[\phi _{0}]h[\phi _{0}]^{*}[0^{t},{\overline{\phi }}_{0}^{t}]^{*}=0\). This means that both of them are eigenvectors of \(h[\phi _{0}]h[\phi _{0}]^{*}\) associated with the eigenvalue 0. This implies that if \(h[\phi _{0}]h[\phi _{0}]^{*}\) has rank \(2m-2\) then \(h[\phi _{0}]h[\phi _{0}]^{*}[\psi _{1,\kappa }^{t},{\overline{\psi }}_{1,\kappa }^{t}]^{t}=0\) if and only if \(\psi _{1,\kappa }=0\). \(\square \)
6.2.3 \(B_{k}+E_{k}-G_{k}+C_{k}=0\) with \(\kappa \le k< 2 \kappa \)
Notice that
and
Therefore,
In particular, when \(k=\kappa =n+1\), since \(\langle \psi _{1,\kappa },\phi _{0}\rangle =0\), one has
This implies that
This implies
Thus,
where
We will combine this with \(B_{1,\kappa }+E_{1,\kappa }-G_{1,\kappa }+C_{1,\kappa }=0\) to solve \(\psi _{1,\kappa }\) and \(\phi _{\kappa }\) in the later subsection.
6.2.4 The case: \(k=\kappa \)
We rewrite (6.33) as follows:
where \({\tilde{\Phi }}_{\kappa -1}\) depends on \(\phi _0,\cdots ,\phi _{\kappa -1}\).
Let U be a \((2m)\times (2m)\) unitary matrix such that
where \(D(\lambda _{1},\cdots ,\lambda _{s},0,\cdots ,0)\) is a diagonal matrix with positive eigenvalues \(\lambda _{1},\cdots ,\lambda _{s}\). Since
we can choose U such that
By (6.29), one has
Let
Therefore, by (6.29) ,
and by (6.28) ,
By (6.38) ,
Then
In general, one can solve \(\psi _{1, \kappa }\) as follows. Let V(z) be the range of \(h[\phi _0] h[\phi _0]^*\) on \({\mathbb {C}}^{2m}\) and let \({\tilde{H}}(z)\) be the projection of \(\left[ \begin{matrix} {\tilde{\Phi }}_{\kappa -1} \\ \overline{{\tilde{\Phi }}}_{\kappa -1}\\ \end{matrix}\right] \) onto V(z). By (6.35) and \(h[\phi _0] h[\phi _0]^* \left[ \begin{matrix} \psi _{1, \kappa } \\ \overline{\psi _{1, \kappa }}\\ \end{matrix}\right] =0\), one can see that
We know from (6.32), \(A_{\kappa , 0}\) can be solved smoothly. Only part of entries of \(\phi _{\kappa }\) are uniquely determined by the equations (6.3). Since
one has
where
Therefore,
Note. We point out here that \(y_{s+1},\cdots ,y_{2m}\) above are with no restrictions. Therefore, \(\phi _{\kappa }\) can not be solved in general from \(B_{\kappa }+C_{\kappa }-G_{\kappa }+E_{\kappa }=0\).
Proposition 6.3
With the notation above, one has
Since the proof of Proposition 6.3 is complicated and very long, we move the proof to Section 7.
6.2.5 \(\kappa _{1}<k<\kappa _{2}-1\)
We start with the following proposition.
Proposition 6.4
Proof
By (6.8) and Proposition 2.3, one has
The proof of the proposition is complete. \(\square \)
Notice that
and
We have
where
In particular,
Case 1. \(k\ne n+2\). One has
and
When \(k=n+2\) and \(\ell =1\), we have
By Proposition 6.3, one has
and
When \(\kappa _{1}<k<\kappa _{2}-1\), by (6.30), one has
In particular,
If \(k\ne n+2\), then
Thus,
and
If \(k=n+2\), by Proposition 6.4, one has \(\langle D_{0},\phi _{0}\rangle \ne 0\). Then
and
By (6.41), we have
From the above equation and the similar argument of subsection 6.1.2, we can solve \(\psi _{1,k}\). By (6.45),
From the above equation and the similar argument of subsection 6.1.2, one can easily solve \(\phi _{k}\).
When \(k\ge \kappa _{2}\), one can solve all \(\phi _{k}\) and \(\psi _{\ell ,k}\) similarly without the complication for the case \(k=n+1\) and \(k=n+2\).
Remark
Let
and
From the expression:
one can see that \(B_{\ell ,k}\left[ u\right] \) depends on \(\phi _{j}\) and \(\psi _{s,\alpha }\in {\mathcal {G}}_{\ell ,k}.\) It is easy to see that \(G_{\ell ,k}\left[ u\right] ,\) \(E_{\ell ,k}\left[ u\right] \), \(C_{\ell ,k}\left[ u\right] \) and therefore, \(X_{\ell ,k}\) depend on \(\phi _{j}\) and \(\psi _{s,\alpha }\in {\mathcal {G}}_{\ell ,k}\).
Assume that \(\kappa _{\ell }\le k<\kappa _{\ell +1}\) and that we have solved \(\left\{ \phi _{j}:j\le k-1\right\} \ \)and \(\left\{ \psi _{s,\alpha } :s\le \ell ,\alpha \le k-1\right\} \) by equations
Then by \(X_{\alpha ,k}=0,\alpha =\ell ,\ell -1,\cdots ,1\), we can solve \(\psi _{\ell ,k},\psi _{\ell -1,k},\cdots , \psi _{1,k} \), respectively and by \(B_{k}+E_{k}-G_{k}+C_{k}=0\) we can solve \(\phi _{k}\). Therefore, if \(\phi _{1} ,\cdots ,\phi _{\kappa },\psi _{1,\kappa },\psi _{1,\kappa +1}, \phi _{\kappa +1}\) are given, then by Equation (6.3), we can compute all \(\left\{ \phi _{k}\right\} \) and \(\left\{ \psi _{\ell ,k}\right\} \) uniquely in the following order:
Now we come back to prove Theorem 1.3. We need to determine \(\phi _{n+1}\) and \(A_{n+2, 0}\) so that they agree with the same quantities determined from the coefficients of u.
As a summary of the above, we have constructed a map v:
such that \(\tau ^{s}[v]\) vanishes on \(\partial B_{n}\) in infinite orders. The construction shows that
Let
where \(\varphi _{z}(w)\) is the Mobius transform for \(B_{n}\) with \(\varphi _{z}(0)=z\) and \(\varphi _{z}(z)=0\). Then \(G\left( z,w\right) \) is the Green’s function for \(\Delta _{B_n}=(1-|z|^{2})L\) on \(B_{n}\). It was proved by M. Stoll, Lemma 6.6 in [27] that the Dirichlet boundary value problem
has the unique solution
Let \(0<\alpha <1\) and let \(\Lambda ^{0,\alpha }(B_{n})\) be the space of all functions f in \(B_{n}\) with
and \(\Lambda ^{k,\alpha }(B_{n})\) denote the space of all functions f in \(B_n\) with
for all \(\alpha +\beta +\gamma +2 \ell \le k\). A map \(u\in \Lambda ^{k,\alpha }(B_{n})\) iff every component of u belongs to \(\Lambda ^{k,\alpha }(B_{n})\).
Let \(r^{m}\Lambda ^{k,\alpha }(B_{n})\) denote the space of all functions (or maps ) f in \(B_{n}\) with
The following theorem was proved by Lee and Melrose [9] (similar to Proposition 4.25 in [9] )
Proposition 6.5
With the notation above, if \((-\Delta _{B_n}+k_n) f\in r^m \Lambda ^{k, \alpha }(B_n)\) or \((-\Delta _{B_n} +k_n) f\in r^m \Lambda ^{k,0}(B_n)\)) then \(f\in r^m \Lambda ^{k+2, \alpha }(B_n)\) or (\(f \in r^m \Lambda ^{k+1,\gamma }(B_n)\) for any \(\gamma \) with \(0<\gamma \le 1/2\)). Here \(0<m< {\frac{1}{2}}(n+\sqrt{ n^2+2k_n})\) and \(k_n>0\).
Let A be any differential operator and let
Let
Then
for any \(k\in {\mathbb {N}}\) and \(0<\alpha <1\). By (6.35), we have
By (6.32), one has
with \({\tilde{H}}_{n+1}\in C^{\infty }({\overline{B}}_{n}\setminus \{0\})\). Since
and
one has
and
Therefore, our computations show that
Let
Since \(\Delta _{B_{n}}=\rho (z)\sum _{j=1}^{n}X_{j}\frac{\partial }{\partial {\overline{z}}_{j}}\), one has
Now since
Therefore,
Thus,
where \(f(z), H(z)\in L^{\infty }(B_{n})\cap C^{\infty }(B_{n}\setminus \{0\})\). Here we take \(1\le k\le n\). By Proposition 6.5, one has \(U_{k}\in r^{n+1-k}\Lambda ^{1,\alpha }(B_{n})\) for any \(\alpha \le 1/2\). This implies that
Thus, \(U_{k}\in r(z)^{n+1-k}\Lambda ^{3,\alpha }({\overline{B}}_{n})\). By iteration, one has that
for any \(m\in {\mathbb {N}}\). Notice that \(U_{k}={\frac{u-V_{n}}{r^{k}}}\), one has
for any \(m\in {\mathbb {N}}\). Let
Then \(u_{n+1}\in \bigcap _{k\in {\mathbb {N}}}\Lambda ^{k,\alpha }(B_{n})\). In particular, \(u_{n+1}|_{\partial B_{n}}\in C^{\infty }(\partial B_{n})\). We let \(\phi _{n+1}(z)=u_{n+1}(z/|z|)\) and let the new \({\tilde{V}}_{n+1}\) be defined as
Repeating the above argument for \(\left\langle u-{\tilde{V}}_{n+1},\phi _{0}\right\rangle ,\) we have
Let
Then we can solve \(\psi _{1,n+2}\) and \(\phi _{n+2}\) and then all the other \(\psi _{j,k}\) and \(\phi _{k}.\) Then one can find v above such that \(u(z)-v(z)\) vanishes on \(\partial B_{n}\) in infinite orders.
Let \(0\le \chi \le 1\), \(\chi (r(z))\in C^{\infty }({\overline{B}}_{n})\) with \(\chi (r(z))=1\) if \(r(z)>-1/2\) and \(\chi (r(z))=0\) if \(r(z)<-3/4\). Then \( \Delta _{B_n}(u-\chi v)\) is smooth in \({\overline{B}}_{n}\) and vanishes on \( \partial B_{n}\) in infinite orders. Thus,
Therefore,
This proves Theorem 1.3. \(\square \)
7 Proof of Proposition 6.3
In this section, we will prove Proposition 6.3.
Proof
Since
We will compute \({\textrm{Im}}\,\langle \Psi _{1,n+1},\phi _{0}\rangle \) term by term.
By (4.23)
where
and
Therefore,
Therefore, By (6.28),
Since
and (6.28), one has
Since
and
we have
Therefore,
Notice that
This implies for \(j=1,\cdots ,n,\) that
and for any real-valued function A,
Therefore,
Notice
and
Then with \(\langle \psi _{1,n+1},\phi _{0}\rangle =0\),
and
Since \({{\mathcal {A}}}_{1}=2\langle \phi _{1},\phi _{0}\rangle \), one has
Therefore,
and
Therefore,
This coupling with (7.7) implies
Now compute \({\textrm{Im}}\,\langle \psi _{1,n+1},n{{\mathcal {A}}}_{1}\phi _{1}\rangle \). Since
by (7.6), one has
Therefore,
Notice that
Therefore,
This implies
Thus,
and
Therefore,
Thus,
Notice that
and
Then
Since
and (7.7), one has
Therefore, by (7.9)
The proof of the proposition is complete. \(\square \)
8 When \(h[\phi _0] h[\phi _0]^*\) has rank \(2m-2\)
When \(h[\phi _0] h[\phi _0]^*\) has rank \(2m-2\), the harmonic map u has better regularity. Precisely, we can state it as the following theorem.
Theorem 8.1
Let \(\phi _{0}:\partial B_{n}\rightarrow \partial B_{m}\) be a \( C^{\infty }\) map satisfying (1.4) and \(h[\phi _{0}]h[\phi _{0}]^{*} \) having the rank \(2m-2\). Then
(a) If \(E_{b}[\phi _{0}]>0\) on \(\partial B_{n}\) then the solution u of (1.8) belongs to \(C^{n+1, \alpha }({\overline{B}}_n)\) for any \(0<\alpha <1\);
(ii) If \(|\partial _b u| |{\overline{\partial }}_b u|> 0\) on \(\partial B_n\), then the solution u of (1.8) has the following asymptotic expansion:
where \(\phi \) and \(\psi _{\ell }\in C^{\infty }({\overline{B}}_{n})\).
Proof
Since \(h[\phi _{0}]h[\phi _{0}]^{*}\) has rank \(2m-2\), one has \( \psi _{1,n+1}=0\). Thus, if \(E_{b}[\phi _{0}]\ne 0\) on \(\partial B_{n}\), one has \(u\in C^{n+1,\alpha }\) for all \(0<\alpha <1\). If \(|\partial _{b}u|| {\overline{\partial }}_{b}u|>0\) on \(\partial B_{n}\), by Theorem 1.3, one has Part (ii) of the theorem holds. \(\square \)
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Dedicate to Professor Peter Li in honor of his 70th birthday.
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The Ren-Yu Chen was supported in part by NSF of China (Grant Nos: 11771323, 12171353).
The Jie Luo was supported in part by PSF of China (Grant No: 2020M680715).
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Chen, RY., Li, SY. & Luo, J. On the Asymptotic Expansions of the Proper Harmonic Maps Between Balls in Bergman Metrics. J Geom Anal 33, 114 (2023). https://doi.org/10.1007/s12220-022-01020-z
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DOI: https://doi.org/10.1007/s12220-022-01020-z