Proof. As before, let \( [ R , \infty ) \times \ensuremath { \mathbb { T } } ^ { 3 } = \{ ( r , \theta ) | r \geq R , \theta = ( \theta _ { 1 } , \theta _ { 2 } , \theta _ { 3 } ) \in \ensuremath { \mathbb { R } } ^ { 3 } / \Lambda \} \). Let \( \Lambda ^ { * } \) be the dual lattice. Choose \( \lambda \in \Lambda ^ { * } - \{ 0 \} \) with minimal length. Choose \( ( a _ { 1 } , a _ { 2 } , a _ { 3 } ) \in \mathbb { S } ^ { 2 } \) such that

\[ ( a _ { 1 } I ^ { * } + a _ { 2 } J ^ { * } + a _ { 3 } K ^ { * } ) \mathrm { d } r = - \frac { \lambda _ { 1 } \mathrm { d } \theta _ { 1 } + \lambda _ { 2 } \mathrm { d } \theta _ { 2 } + \lambda _ { 3 } \mathrm { d } \theta _ { 3 } } { | \lambda | } . \]

Use \( a _ { 1 } I \! + \! a _ { 2 } J \! + \! a _ { 3 } K \) as the complex structure. Then the function \( e ^ { 2 \pi | \lambda | r } e ^ { 2 \pi i < \lambda , \theta > } \) is holomorphic. The growth rate of this function is exactly \( O ( e ^ { 2 \pi | \lambda | r } ) \).

Now we pull back this function from \( [ R , \infty ) \times \mathbb { T } ^ { 3 } \) to M, cut it off and fill in with 0 inside K, we obtain a function f satisfying

\[ \bar { \partial } _ { g } f = \phi = { \cal O } ( e ^ { ( 2 \pi | \lambda | - \mu ) r } ) , \]

where \( \mu \) is the constant in Theorem 4.17. So \( \phi \in \underline { { L } } _ { - 2 \delta } ^ { 2 } \) for any non-critical positive number \( \delta \in ( 2 \pi | \lambda | - \mu , 2 \pi | \lambda | ) \). By Theorem 4.20, there exists \( \psi \in \underline { { H ^ { 2 } } } _ { 2 \delta } \) satisfying

\[ \phi = \Delta \psi = - ( { \bar { \partial } } ^ { * } { \bar { \partial } } + { \bar { \partial } } { \bar { \partial } } ^ { * } ) \psi \]

in the distribution sense. Elliptic regularity implies that \( \psi \) is a smooth (0, 1)-form. As before, \( \bar { \partial } \psi = \xi \omega ^ { + } \) is a harmonic (0,2)-form. So \( \xi \) is a harmonic function of order \( O ( e ^ { \delta r } ) \).

Now we use a cut-off function and the diffeomorphism to average g and the pull back of h. We get a smooth metric \( g ^ { \prime } \) on M which is identically the pull back of h outside a very big ball. Now let \( \nu \) be the inferior of positive \( \nu ^ { \prime } \) such that \( \xi \) is bounded by \( O ( e ^ { \nu ^ { \prime } r } ) \). If \( \nu > \) 0, then \( \Delta _ { g ^ { \prime } } \xi \, \in \, \underline { { L } } _ { - 2 \delta } ^ { 2 } \) 0 for any positive \( \nu > \delta ^ { \prime } > \nu - \mu \). It follows that there exists a function in \( \underline { { L } } _ { - 2 \delta ^ { \prime } } ^ { 2 } \) whose Laplacian \( \Delta _ { g ^ { \prime } } \) is \( \Delta _ { g ^ { \prime } } \xi \). The difference of those two functions is a \( g ^ { \prime } . \)-harmonic function. By Lemma 4.21, it must have at most linear growth rate since the growth rate is below the first nonlinear harmonic function. It follows that \( \xi \) must lie in \( O ( e ^ { \delta ^ { \prime } r } ) \), a contradiction. So \( \nu \) = 0. Therefore, \( \xi \) is bounded by any exponential growth function.

So \( \Delta _ { g ^ { \prime } } \xi \) decay exponentially. In particular, it’s in \( L _ { 1 - \epsilon } ^ { 2 } \). By Theorem 4.12, we can find out a function in \( H _ { - 3 - \epsilon } ^ { 2 } \) whose \( \Delta _ { g ^ { \prime } } \) is \( \Delta _ { g ^ { \prime } } \xi \). Therefore, we know that \( \xi \) is actually in \( O ( r ^ { 1 + \epsilon } ) \). Of course, \( \bar { \partial } \psi = \xi \omega ^ { + } \) has the same estimate.

By Lemma 4.4, the harmonic (0,1)-form \( \bar { \partial } ^ { \ast } \bar { \partial } \psi = \bar { \partial } ( f + \bar { \partial } ^ { \ast } \psi ) \) is in \( O ( r ^ { \epsilon } ) \)) and its covariant derivative is in \( O ( r ^ { - 1 + \epsilon } ) \). The Weitzenb¨ock formula implies that \( \nabla ^ { * } \nabla ( \bar { \partial } ^ { * } \bar { \partial } \psi ) = 0 \). Therefore

\[ \int _ { M } | \nabla ( \bar { \partial } ^ { * } \bar { \partial } \psi ) | ^ { 2 } \chi \leq \int _ { M } | \bar { \partial } ^ { * } \bar { \partial } \psi | | \nabla ( \bar { \partial } ^ { * } \bar { \partial } \psi ) | | \nabla \chi | \]

for any smooth compactly supported \( \chi \). Let \( \chi = \chi ( r \! - \! R ) \), the right hand side converges to 0. Therefore \( \partial ^ { * } \bar { \partial } \psi \) is a covariant constant (0,1)-form. If this form is non-zero, it would be invariant under the holonomy of any loop. However, elements in \( \mathrm { S U } \)\( 2 \)) have no fixed point except the identity matrix. So \( ( M , g ) \) must have trivial holonomy. Therefore, it’s \( \mathbb { R } ^ { 4 - k } \times \mathbb { T } ^ { k } \) with flat metric. It’s a