4.2. Asymptotic for the first order ansatz I [044X]

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4.2. Asymptotic for the first order ansatz I

The following two Sections study the leading order behaviour of the Kähler ansatz away from $S$ at large distance. The region under consideration lies over

(4.14)

\begin{split}\{y_{2}>1,A^{1/4}|\mu|+|y_{1}|>e^{-2\pi y_{2}}\}\cup\{y_{1}>1,A^{1/4}|\mu|+|y_{2}|>e^{-2\pi y_{1}}\}\\ \cup\{y_{1}<-1,A^{1/4}|\mu|+|y_{2}-y_{1}|>e^{2\pi y_{1}}\}\cup\{|\mu|>A^{-1/4}\}.\end{split}

This is a quantitative way of asserting boundedness away from $S$ .

We define the average functions of $\gamma_{i}$ (cf. (4.11)) on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$ by

(4.15)

\bar{\gamma}_{i}(y_{1},y_{2},\mu)=\int_{0}^{1}\int_{0}^{1}\gamma_{i}(x_{1}+\sqrt{-1}y_{1},x_{2}+\sqrt{-1}y_{2},\mu)dx_{1}dx_{2}.

This Section is concerned with describing the behaviour of $\bar{\gamma}_{i}$ , and next Section proves exponential decay estimate for $|\gamma_{i}-\bar{\gamma}_{i}|$ .

Recall from (4.3) the Euclidean metric $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$ . Its volume measure is $d\text{Vol}_{a}^{\prime}=\frac{A^{3/2}}{\sqrt{\mathbb{A}}}dy_{1}dy_{2}d\mu,$ and the associated Laplacian is $\Delta_{a}^{\prime}=a^{p\bar{q}}\frac{\partial^{2}}{\partial y_{p}\partial y_{q}}+A^{-1}\frac{\partial^{2}}{\partial\mu\partial\mu}$ . Here some care is needed in the calculations regarding the difference between Hermitian and symmetric matrices.

Lemma 4.8.

(Harmonicity) In the region (4.14) inside $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{\mu}$ the functions $\bar{\gamma}_{i}$ satisfy $\Delta_{a}^{\prime}\bar{\gamma}_{i}=0$ , or equivalently their pullbacks to $(\mathbb{C}^{*})^{2}_{\eta_{1},\eta_{2}}\times\mathbb{R}_{\mu}$ satisfy $\Delta_{a}\bar{\gamma}_{i}=0$ .

Proof.

The amoeba $\text{Im}(S)$ is disjoint from the region (4.14), so Proposition 4.4 asserts the $\Delta_{a}$ -harmonicity of $\gamma_{i}$ , whence the harmonicty of $\bar{\gamma}_{i}$ . ∎

Next we wish to write $\bar{\gamma}_{i}$ also in terms of a Green’s representation. From the calculation in Lemma 4.1,

(4.16)

\bar{\gamma}=\int_{0}^{1}\int_{0}^{1}\gamma(x_{1}+\sqrt{-1}y_{1},x_{2}+\sqrt{-1}y_{2},\mu)dx_{1}dx_{2}=-\frac{1}{4\pi\varrho\sqrt{\mathbb{A}}}.

Thus by integrating (4.11) in the $x_{1},x_{2}$ variables,

Corollary 4.9.

(Green’s representation formula for $\bar{\gamma}_{i}$ )

\begin{cases}\bar{\gamma}_{1}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\gamma}_{2}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{1}^{\prime}<\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{1}^{\prime}\wedge(d\bar{\eta}_{1}^{\prime}-d\bar{\eta}_{2}^{\prime})-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log 2\Lambda\}\\ \bar{\gamma}_{3}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{1}^{\prime}>-\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\gamma}_{4}(y_{1},y_{2},\mu)=&\text{Im}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}\}.\end{cases}

Our goal is to extract the leading order behvaiour in terms of an explicit elementary formula. For this purpose we essentially replace $S$ by its asymptotic cylinders $\mathfrak{D}_{i}\times S^{1}$ . Define

\begin{cases}\bar{\bar{\gamma}}_{1}(y_{1},y_{2},\mu)=&\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{2\sqrt{\mathbb{A}}}\int_{0}^{\Lambda}|(y_{1},y_{2}-s,\mu)|_{a}^{\prime-1}ds-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\bar{\gamma}}_{2}(y_{1},y_{2},\mu)=&\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{2\sqrt{\mathbb{A}}}\int_{0}^{\Lambda}|(y_{1}-s,y_{2},\mu)|_{a}^{\prime-1}ds-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log 2\Lambda\}\\ \bar{\bar{\gamma}}_{3}(y_{1},y_{2},\mu)=&\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{2\sqrt{\mathbb{A}}}\int_{-\Lambda}^{0}|(y_{1}-s,y_{2}-s,\mu)|_{a}^{\prime-1}ds\\ &-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\log 2\Lambda\}.\end{cases}

By construction $\bar{\bar{\gamma}}_{i}$ is $\Delta_{a}^{\prime}$ -harmonic in the region (4.14). Morever,

Lemma 4.10.

(Estimate of remainder terms) In the region (4.14) we have $|\bar{\gamma}_{4}|\leq C\varrho^{-1}$ . For $i=1,2,3$ , in the subset of (4.14) where $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{i})\gtrsim A^{1/4}$ we have $|\bar{\gamma}_{i}-\bar{\bar{\gamma}}_{i}|\leq C\varrho^{-1}$ .

Proof.

We use the Green representation of $\bar{\gamma}_{4}$ . The total measure

\int_{S}\text{Im}(\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime})\leq C,

so the contribution to $\bar{\gamma}_{4}$ from the ball $\{|(\eta_{1}^{\prime},\eta_{2}^{\prime},0)|_{a}\lesssim A^{1/4}\}\subset S$ is bounded by $C\varrho^{-1}$ . The contributions from the 3 ends are neglegible unless the point $(y_{1},y_{2},\mu)$ inside the region (4.14) is close to $S$ along some $\mathfrak{D}_{i}$ ; we focus on the case of $\mathfrak{D}_{1}$ . The key fact is the exponential decay of the measure: along $\mathfrak{D}_{1}$ we have

\text{Im}(\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime})\leq Ce^{-2\pi y_{2}^{\prime}}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{2}^{\prime}.

Thus the contribution from the end $\{y_{2}^{\prime}>1\}\cap S$ is controlled by

\begin{split}C\int_{0}^{\infty}e^{-2\pi y_{2}^{\prime}}|(y_{1},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}dy_{2}^{\prime}\leq C\varrho^{-1}.\end{split}

which implies the estimates on $\bar{\gamma}_{4}$ .

For $\bar{\gamma}_{i}-\bar{\bar{\gamma}}_{i}$ , the main point is that $S$ approaches its asymptotic cylinder at an exponentially fast rate. The rest of the arguments are similar. ∎

Elementary integration gives

Lemma 4.11.

(Leading order asymptote) The formulae for $\bar{\bar{\gamma}}_{i}$ are given explicitly as

(4.17)

\begin{cases}\bar{\bar{\gamma}}_{1}=&-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log(\frac{\mathbb{A}^{1/2}}{\sqrt{Aa_{2\bar{2}}}}|(y_{1},y_{2},\mu)|_{a}^{\prime}-y_{2}-\frac{\text{Re}(a_{1\bar{2}})}{a_{2\bar{2}}}y_{1})\\ \bar{\bar{\gamma}}_{2}=&-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log(\frac{\mathbb{A}^{1/2}}{\sqrt{Aa_{1\bar{1}}}}|(y_{1},y_{2},\mu)|_{a}^{\prime}-y_{1}-\frac{\text{Re}(a_{1\bar{2}})}{a_{1\bar{1}}}y_{2})\\ \bar{\bar{\gamma}}_{3}=&-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\\ &\log(\frac{\mathbb{A}^{1/2}}{\sqrt{A(a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}})}}|(y_{1},y_{2},\mu)|_{a}^{\prime}+\frac{a_{1\bar{1}}y_{1}+\text{Re}(a_{1\bar{2}})y_{2}+\text{Re}(a_{2\bar{1}})y_{1}+a_{2\bar{2}}y_{2}}{a_{1\bar{1}}+2\text{Re}(a_{1\bar{2}})+a_{2\bar{2}}}).\end{cases}

Remark 4.3.

These formulae have strong similarity with $\bar{\alpha}_{i}$ in (3.6) except for the absence of an additive constant as in (3.6), which is an artefact of a non-canonical choice of constant in our definition of $\gamma_{i}$ (cf. Remark 4.2).