ScalingStacks

7.1.3. Subdivision of the manifold ℳ [03J0]

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7.1.3. Subdivision of the manifold ℳ\mathcal{M}

Fix a gluing parameter β>1\beta>1, the manifold (ℳ,gβ)(\mathcal{M},g_{\beta}) will be divided into the following 99 regions depending on the different collapsing behaviors of metric gg:

I:dm​(𝒙)≤β−12​for some​ 1≤m≤m0\displaystyle\I:\ d_{m}(\bm{x})\leq\beta^{-\frac{1}{2}}\ \text{for some}\ 1\leq m\leq m_{0}
    (in the neck, very close to a monopole point)
II: 2​β−12≤dm​(𝒙)≤ι0′4⋅(β)12​ for some​ 1≤m≤m0\displaystyle\II:\ 2\beta^{-\frac{1}{2}}\leq d_{m}(\bm{x})\leq\frac{\iota_{0}^{\prime}}{4}\cdot(\beta)^{\frac{1}{2}}\text{ for some}\ 1\leq m\leq m_{0}
    (in the neck, not close, but not too far from any monopole point)
III:z⁡(𝒙)∈[−m0​T0,m0​T0]​and ​dm​(𝒙)≥ι0′2⋅(β)12​ for all​ 1≤m≤m0\displaystyle\III:\ z(\bm{x})\in[-m_{0}T_{0},m_{0}T_{0}]\ \text{and }d_{m}(\bm{x})\geq\frac{\iota_{0}^{\prime}}{2}\cdot(\beta)^{\frac{1}{2}}\text{ for all}\ 1\leq m\leq m_{0}
    (in a bounded region of the neck, but far from any monopole point)
IV−:z(𝒙)∈[−T−/2,−2m0T0]\displaystyle\IV_{-}:\ z(\bm{x})\in[-T_{-}/2,-2m_{0}T_{0}]
    (in the negative end region of the neck)
IV+:z⁡(𝒙)∈[2​m0​T0,T+/2]\displaystyle\IV_{+}:\ z(\bm{x})\in[2m_{0}T_{0},T_{+}/2]
    (in the positive end region of the neck)
V−:𝒙∈Xb−4​and​ 2​ζ0−≤z−​(𝒙)≤T−\displaystyle\V_{-}:\ \bm{x}\in X_{b_{-}}^{4}\ \text{and}\ 2\zeta_{0}^{-}\leq z_{-}(\bm{x})\leq T_{-}
    (in the end region of Xb−4X_{b_{-}}^{4})
V+:𝒙∈Xb+4​and​ 2​ζ0+≤z+​(𝒙)≤T+\displaystyle\V_{+}:\ \bm{x}\in X_{b_{+}}^{4}\ \text{and}\ 2\zeta_{0}^{+}\leq z_{+}(\bm{x})\leq T_{+}
    (in the end region of Xb+X_{b_{+}})
VI−:𝒙∈BD0−​(q−)¯⊂Xb−4\displaystyle\VI_{-}:\ \bm{x}\in\overline{B_{D_{0}^{-}}(q_{-})}\subset X_{b_{-}}^{4}
    (in the bounded part of Xb−4X_{b_{-}}^{4})
VI+:𝒙∈BD0+​(q+)¯⊂Xb+4\displaystyle\VI_{+}:\ \bm{x}\in\overline{B_{D_{0}^{+}}(q_{+})}\subset X_{b_{+}}^{4}
(in the bounded part of Xb+4).\displaystyle\hskip 28.45274pt\mbox{(in the bounded part of $X_{b_{+}}^{4}$)}.

We note that for 𝒙∈IV±\bm{x}\in\IV_{\pm}, we have

(7.18) T0′​(β)12≤dm​(𝒙)≤R±​ for all​ 1≤m≤m0,\displaystyle T_{0}^{\prime}(\beta)^{\frac{1}{2}}\leq d_{m}(\bm{x})\leq R_{\pm}\text{ for all}\ 1\leq m\leq m_{0},

where

(7.19) R−\displaystyle R_{-} ≡sup{dg​(x,p1)|−T−≤z⁡(𝒙)≤0},\displaystyle\equiv\sup\Big\{d_{g}(x,p_{1})\Big|-T_{-}\leq z(\bm{x})\leq 0\Big\},
(7.20) R+\displaystyle R_{+} ≡sup{dg​(x,p1)|0≤z⁡(𝒙)≤T+}.\displaystyle\equiv\sup\Big\{d_{g}(x,p_{1})\Big|0\leq z(\bm{x})\leq T_{+}\Big\}.

Immediately, there is some constant C>0C>0 (independent of β\beta) such that

(7.21) C−1​β32≤R±≤C​β32.C^{-1}\beta^{\frac{3}{2}}\leq R_{\pm}\leq C\beta^{\frac{3}{2}}.
Remark 7.1.

Notice that the above regions do not completely cover the manifold ℳ\mathcal{M}. However, each gap region shares the geometric behavior with the adjacent regions in the above subdivision. Therefore, the curvature estimates and the rescaled geometries in each gap region will be the same as in the adjacent regions, so we will ignore these gap regions in the following.

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