5.4. Asymptotics of harmonic functions on the Calabi model space [054D]

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5.4. Asymptotics of harmonic functions on the Calabi model space

As Section 5.1, we fix $r_{0}\in(0,1)$ , and view the Calabi model space $\mathcal{C}^{n}$ as the product of a fixed cross section $Y^{2n-1}\cong\{\rho=\rho_{0}\}$ with the restricted metric $h_{0}=g_{\mathcal{C}^{n}}|_{\{\rho=\rho_{0}\}}$ with a ray $\mathbb{R}^{+}$ . The spectrum of the Laplacian operator on $Y$ is given by $\{\Lambda_{k}\}_{k=0}^{\infty}$ , with $\Lambda_{0}=0$ , and we have chosen an orthonormal basis of complex valued eigenfunctions of the form $\{\varphi_{k}\}_{k=0}^{\infty}$ such that

(5.175)

\displaystyle\begin{cases}\Delta_{Y^{2n-1}}\varphi_{k}=\Lambda_{k}\cdot\varphi_{k},\\ \|\varphi_{k}\|_{L^{2}(Y^{2n-1})}=1.\end{cases}

We need a basic lemma on the decay of Fourier coefficients of the expansion of a sufficiently smooth function in terms of eigenfunctions.

Lemma 5.13.

Let $K_{0}\geq 1$ and let $\xi\in C^{2K_{0}}(Y^{2n-1})$ satisfy the $L^{2}$ -expansion

(5.176)

\xi(\bm{y})=\sum\limits_{k=1}^{\infty}\xi_{k}\cdot\varphi_{k}(\bm{y}),

then for all $k\in\mathbb{Z}_{+}$ ,

(5.177)

|\xi_{k}|\leq\frac{C|\xi|_{C^{2K_{0}}(Y^{2n-1})}}{(\Lambda_{k})^{K_{0}}},

where the constant $C>0$ is independent of $k$ .

Proof.

The estimate is proved by the standard integration by parts. Since the eigenfunctions $\varphi_{k}$ satisfy

(5.178)

-\Delta_{h_{0}}\varphi_{k}=\Lambda_{k}\cdot\varphi_{k}

and $\|\varphi_{k}\|_{L^{2}(Y^{2n-1})}=1$ , we have that

	$\displaystyle\|\xi_{k}(z)\|$	$\displaystyle=\Big\|\int_{Y^{2n-1}}\xi\cdot\varphi_{k}\Big\|=\Big\|\int_{Y^{2n-1}}\xi\cdot\frac{(-\Delta_{h_{0}})^{K_{0}}\varphi_{k}}{(\Lambda_{k})^{K_{0}}}\Big\|$
(5.179)			$\displaystyle\leq\frac{1}{(\Lambda_{k})^{K_{0}}}\int_{Y^{2n-1}}\|\Delta_{h_{0}}^{K_{0}}\xi\|\cdot\|\varphi_{k}\|$
(5.180)			$\displaystyle\leq\frac{C\|\xi\|_{C^{2K_{0}}(Y^{2n-1})}}{(\Lambda_{k})^{K_{0}}},$

where $C>0$ depends only on the geometry of $Y^{2n-1}$ .

∎

Proposition 5.14 (Asymptotics of harmonic functions).

Let $(\mathcal{C}^{n},g_{\mathcal{C}^{n}})$ be a Calabi model space with $\dim_{\mathbb{C}}(\mathcal{C}^{n})=n$ . Define a constant

(5.181)

\delta_{b}\equiv 2\underline{\lambda}^{\frac{1}{2}}n^{-\frac{1}{2}}>0

where $\underline{\lambda}>0$ is given by (5.14). If $u$ is a harmonic function outside a compact set in $\mathcal{C}^{n}$ satisfying

(5.182)

|u(z,\bm{y})|=O(e^{\delta\cdot z^{\frac{n}{2}}})

for some $\delta\in(0,\delta_{b})$ as $z\rightarrow\infty$ . Then $u$ can be decomposed as

(5.183)

u(z,\bm{y})=L(z)+h(z,\bm{y})

with the following properties:

(1)

$L(z)=\kappa_{0}\cdot z+c_{0}$ for some $\kappa_{0},c_{0}\in\mathbb{R}$ .
(2)

$h(z,\bm{y})$ is harmonic and for any $k\in\mathbb{N}$ , there is some $C_{k}>0$ such that

(5.184) $|\nabla^{k}h(z,\bm{y})|\leq C_{k}\cdot e^{-\underline{\delta}\cdot z^{\frac{n}{2}}}$

for all $\underline{\delta}\in(0,\delta_{b})$ , as $z\to+\infty$ .

Proof.

The proof consists of two steps.

In the first step, we will apply separation of variables to show that if a harmonic function $u$ satisfies (5.182), then $u(z,\bm{y})=k_{0}\cdot z+c_{0}+h(z,\bm{y})$ for some $k_{0},c_{0}\in\mathbb{R}$ and $h(z,\bm{y})$ has some exponential decaying rate.

Since $u$ is smooth, for any fixed $z\geq 1$ , we have the fiber-wise $L^{2}$ -expansion of $u$ as follows,

(5.185)

u(z,\bm{y})=\sum\limits_{k=1}^{\infty}u_{k}(z)\cdot\varphi_{k}(\bm{y}),

where $\bm{y}\in Y^{2n-1}$ and $u_{k}$ satisfies the equation

(5.186)

\frac{d^{2}u_{k}(z)}{dz^{2}}-(\frac{j_{k}^{2}n^{2}}{4}\cdot z^{n}+n\lambda_{k})z^{n-2}u_{k}(z)=0,\ z\geq 1,

for some $j_{k}\in\mathbb{N}$ and $\lambda_{k}\geq 0$ . Notice that the expansion (5.185) converges in the $C^{\infty}$ -topology. This follows from Lemma 5.13, Lemma 3.32 and the Weyl law for spectrum asymptotics.

For $k=0$ we have $j_{k}=\lambda_{k}=0$ , and $u_{k}$ is a linear function of the form $\kappa_{0}\cdot z+c_{0}$ . For $k\geq 1$ , we can write $u_{k}$ as a linear combination of the two linearly independent solutions discussed in Section 5.2 and 5.3.

(5.187)

u_{k}(z)=C_{k}\cdot\mathcal{D}_{k}(z)+C_{k}^{*}\cdot\mathcal{G}_{k}(z),

where $\mathcal{G}_{k}$ is a growing and $\mathcal{D}_{k}$ is decaying.

We claim $C_{k}^{*}=0$ for all $k\in\mathbb{Z}_{+}$ . To see this, we apply Lemma 5.13 to $u(z,\bm{y})$ , then for all $k\in\mathbb{Z}_{+}$

(5.188)

|u_{k}(z)|=O(e^{\delta z^{\frac{n}{2}}}).

So the claim follows from the asymptotics of $\mathcal{G}_{k}(z)$ in Lemma 5.4 and 5.7 which corresponds to $j_{k}=0$ and $j_{k}\in\mathbb{Z}_{+}$ respectively.

Now we define

(5.189)

h(z,\bm{y})\equiv u(z,\bm{y})-(\kappa_{0}\cdot z+c_{0})=\sum_{k=1}^{\infty}u_{k}(z)\cdot\varphi_{k}(\bm{y}).

It suffices to show $h(z,\bm{y})$ decays at the desired rate. Let $z_{0}>(2\delta_{b})^{2/n}$ be sufficiently big so that $u$ is defined on $\{z\geq z_{0}\}$ . Now we fix $K_{0}\equiv 2n+1$ . Applying Lemma 5.13 to $u(z_{0},\bm{y})$ we get for all $k\in\mathbb{Z}_{+}$ ,

(5.190)

|u_{k}(z_{0})|\leq C_{1}(\Lambda_{k})^{-K_{0}}.

We separate in several cases. First, we consider $k\in\mathbb{Z}_{+}$ with $j_{k}=0$ . Applying (5.74), then for any $\epsilon>0$ with $\underline{\delta}=(1-\epsilon)\delta_{b}<\delta_{b}$ , if $z\geq\frac{1}{\epsilon^{\frac{2}{n}}}\cdot z_{0}$ ,

(5.191)

\Big|\frac{u_{k}(z)}{u_{k}(z_{0})}\Big|=\Big|\frac{\mathcal{D}_{k}(z)}{\mathcal{D}_{k}(z_{0})}\Big|\leq Ce^{-2\sqrt{\frac{\lambda_{k}}{n}}\cdot(z^{\frac{n}{2}}-z_{0}^{\frac{n}{2}})}\leq Ce^{-(1-\epsilon)\delta_{b}\cdot z^{\frac{n}{2}}}=Ce^{-\underline{\delta}\cdot z^{\frac{n}{2}}}.

This implies that

(5.192)		$\displaystyle\Big\|\sum_{\begin{subarray}{c}k>0\\ j_{k}=0\end{subarray}}\frac{u_{k}(z)}{u_{k}(z_{0})}\cdot u_{k}(z_{0})\cdot\varphi_{k}(\bm{y})\Big\|$	$\displaystyle\leq$	$\displaystyle\sum_{\begin{subarray}{c}k>0\\ j_{k}=0\end{subarray}}\Big\|\frac{u_{k}(z)}{u_{k}(z_{0})}\Big\|\cdot\|u_{k}(z_{0})\|\cdot\|\varphi_{k}(\bm{y})\|$
(5.192)			$\displaystyle\leq$	$\displaystyle Ce^{-\delta_{b}\cdot z^{\frac{n}{2}}}\cdot\sum_{\begin{subarray}{c}k>0\\ j_{k}=0\end{subarray}}\frac{1}{(\Lambda_{k})^{K_{0}-\frac{n}{2}}},$

where the eigenfunction estimate

(5.193)

\|\varphi_{k}\|_{L^{\infty}(Y^{2n-1})}\leq C\cdot(\Lambda_{k})^{\frac{n}{2}}.

follows from Lemma 3.32.

When $j_{k}\in\mathbb{Z}_{+}$ we divide into two cases. When $Q\geq 1$ we apply Corollary 5.9.1 and Lemma 5.11 (with $\eta=2\delta_{b}$ ) to get

(5.194)		$\displaystyle\Big\|\sum_{\begin{subarray}{c}j_{k}\geq 1\\ Q\geq 1\end{subarray}}\frac{u_{k}(z)}{u_{k}(z_{0})}\cdot u_{k}(z_{0})\cdot\varphi_{k}(\bm{y})\Big\|$	$\displaystyle\leq$	$\displaystyle\sum_{\begin{subarray}{c}j_{k}\geq 1\\ Q\geq 1\end{subarray}}\Big\|\frac{u_{k}(z)}{u_{k}(z_{0})}\Big\|\cdot\|u_{k}(z_{0})\|\cdot\|\varphi_{k}(\bm{y})\|$
(5.194)			$\displaystyle\leq$	$\displaystyle Ce^{-\delta_{b}\cdot z^{\frac{n}{2}}}\sum_{\begin{subarray}{c}j_{k}\geq 1\\ Q\geq 1\end{subarray}}\frac{1}{(\Lambda_{k})^{K_{0}-\frac{n}{2}-1}}.$

Now when $Q\leq 1$ we apply instead Corollary 5.12.1 to get

(5.195)		$\displaystyle\Big\|\sum_{\begin{subarray}{c}j_{k}\geq 1\\ Q\leq 1\end{subarray}}\frac{u_{k}(z)}{u_{k}(z_{0})}\cdot u_{k}(z_{0})\cdot\varphi_{k}(\bm{y})\Big\|$	$\displaystyle\leq$	$\displaystyle\sum_{\begin{subarray}{c}j_{k}\geq 1\\ Q\leq 1\end{subarray}}\Big\|\frac{u_{k}(z)}{u_{k}(z_{0})}\Big\|\cdot\|u_{k}(z_{0})\|\cdot\|\varphi_{k}(\bm{y})\|$
(5.195)			$\displaystyle\leq$	$\displaystyle Ce^{-\frac{z^{n}}{2}}\sum_{\begin{subarray}{c}j_{k}\geq 1\\ Q\leq 1\end{subarray}}\frac{1}{(\Lambda_{k})^{K_{0}-\frac{n}{2}-1}}.$

Summing up all the above we get

(5.196)

|h(z,\bm{y})|\leq Ce^{-\delta_{b}\cdot z^{\frac{n}{2}}}\sum_{k=1}^{\infty}\frac{1}{(\Lambda_{k})^{K_{0}-\frac{n}{2}-1}}.

Since $K_{0}=2n+1$ we see the series converges. So the proof of the first step is done.

The second step is to prove the higher decaying estimate for the error function $h(z,\bm{y})$ , which follows from the uniform Schauder estimate. We have proved that the error function $h(z,\bm{y})$ as a harmonic function satisfies

(5.197)

|h(z,\bm{y})|\leq C_{0}\cdot e^{-\underline{\delta}\cdot z^{\frac{n}{2}}}.

By explicit and straightforward computations, a Calabi space $(\mathcal{C}^{n},g_{\mathcal{C}^{n}})$ is collapsing with bounded curvatures as $z\to+\infty$ . We just lift the harmonic function $h$ to the local universal cover which is non-collapsed with uniformly bounded geometry. So the following Schauder estimate holds for any $k\in\mathbb{Z}_{+}$ and $\alpha\in(0,1)$ on the local universal cover,

(5.198)

|h|_{C^{k,\alpha}(B_{r_{0}}(\bm{x}))}\leq C_{k}\cdot|h|_{C^{0}(B_{2r_{0}}(\bm{x}))}\leq C_{k}\cdot e^{-\underline{\delta}\cdot z^{\frac{n}{2}}}.

where $r_{0}>0$ is some fixed constant of some definite size which is independent of $\bm{x}\in\mathcal{C}^{n}$ . In particular, at the center $\bm{x}=(z,\bm{y})$ , we have

(5.199)

|\nabla^{k}h(z,\bm{y})|\leq C_{k}\cdot|h|_{C^{0}(B_{2r_{0}}(\bm{x}))}\leq C_{k}\cdot e^{-\underline{\delta}\cdot z^{\frac{n}{2}}}.

This completes the proof of (5.184).

∎

	$\displaystyle\|\xi_{k}(z)\|$	$\displaystyle=\Big\|\int_{Y^{2n-1}}\xi\cdot\varphi_{k}\Big\|=\Big\|\int_{Y^{2n-1}}\xi\cdot\frac{(-\Delta_{h_{0}})^{K_{0}}\varphi_{k}}{(\Lambda_{k})^{K_{0}}}\Big\|$
(5.179)			$\displaystyle\leq\frac{1}{(\Lambda_{k})^{K_{0}}}\int_{Y^{2n-1}}\|\Delta_{h_{0}}^{K_{0}}\xi\|\cdot\|\varphi_{k}\|$
(5.180)			$\displaystyle\leq\frac{C\|\xi\|_{C^{2K_{0}}(Y^{2n-1})}}{(\Lambda_{k})^{K_{0}}},$