ScalingStacks

Example 2.12 . [048X]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Example 2.12.

Consider two almost calibrated Lagrangians L1,L2L^{1},L^{2} with a unique intersection point pp, and there is a Darboux chart around pp modelled on B1⊂ℂnB_{1}\subset\mathbb{C}^{n}, such that inside the chart L1,L2L^{1},L^{2} the local setup is

L1=ei​π/n​ℝn,L2=ℝn,ω=−12​∑d​zk∧d​z¯k,Ω≈∏d​zk.L^{1}=e^{i\pi/n}\mathbb{R}^{n},\quad L^{2}=\mathbb{R}^{n},\quad\omega=\frac{\sqrt{-1}}{2}\sum dz_{k}\wedge d\bar{z}_{k},\quad\Omega\approx\prod dz_{k}.

Let (Lt)0<t≪1(L_{t})_{0<t\ll 1} be a 1-parameter family of Lagrangian connected sums with neck length O⁡(t)O(t), which all agree with L1∪L2L_{1}\cup L_{2} except in a compact subset in B1B_{1}. Inside B1B_{1}, we take the ansatz

Lt={(t​γ​(s)​x1,…,t​γ​(s)​xn)|x12+…​xn2=1},L_{t}=\{(t\gamma(s)x_{1},\ldots,t\gamma(s)x_{n})|x_{1}^{2}+\ldots x_{n}^{2}=1\},

where the curve γ⁡(s):ℝ→ℂ\gamma(s):\mathbb{R}\to\mathbb{C} can be chosen so that LtL_{t} is almost calibrated and agrees with L1∪L2L_{1}\cup L_{2} outside B1/2B_{1/2}. Clearly, LtL_{t} are related by scaling inside B1B_{1}. The Hamiltonian vector field along LtL_{t}, which is really a section of (T​X/T​Lt)|Lt(TX/TL_{t})|_{L_{t}}, agrees with (γ⁡(s)​x1,…,γ⁡(s)​xn)(\gamma(s)x_{1},\ldots,\gamma(s)x_{n}) inside B1B_{1} and is zero outside.3838 38 This is consistent because near the boundary of B1B_{1}, the position vector is a tangent vector of LL, so vanishes in the quotient T​X/T​LtTX/TL_{t}. The corresponding Hamiltonian function hth_{t} is t2t^{2} times a smooth function of one variable ss; a small caveat is that hth_{t} converges to two generally different constants along L1L_{1} and L2L_{2}. Thus it takes finite distance in the Solomon metric to reach the limit t→0t\to 0, and the Solomon functional remains finite, but the topology changes from LtL_{t} to L1∪L2L^{1}\cup L^{2}.

Original mathematics by the credited authors. Source-backed reader collection; mathematical self-containment is not assessed.