ScalingStacks

Problem 3.12 . [03P1]

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Problem 3.12.

(a) Choose from the literature your favourite family of explicit, nonsingular, exact SL mm-folds NsN_{s} in ℂm{\mathbin{\mathbb{C}}}^{m} which converge to an explicit singular SL mm-fold N0N_{0} as s→0s\rightarrow 0. For example, let NN be an exact AC SL mm-fold in ℂm{\mathbin{\mathbb{C}}}^{m} with cone C,C, and take Ns=s⋅NN_{s}=s\cdot N for s>0s>0 and N0=CN_{0}=C.

Construct examples {Lt:t∈[0,T]}\{L^{t}:t\in[0,T]\} of Lagrangian MCF in ℂm{\mathbin{\mathbb{C}}}^{m} or in a Calabi–Yau mm-fold (M,J,g,Ω)(M,J,g,\Omega) with finite time singularities at t=Tt=T for which LTL^{T} has a singularity at x∈ℂmx\in{\mathbin{\mathbb{C}}}^{m} modelled on N0,N_{0}, and LtL^{t} near xx for t∈(T−ϵ,T)t\in(T-\epsilon,T) approximates Ns⁡(t),N_{s(t)}, where s⁡(t)→0s(t)\rightarrow 0 as t→T,t\rightarrow T, as in Principle 3.9(a).

(b) If you can do (a), determine whether Lagrangian MCF starting from a small generic Hamiltonian perturbation of L0L^{0} also develops finite time singularities of the same type. In this case, we call this type a generic singularity of Lagrangian MCF. If it is not generic, compute the expected codimension amongst Hamiltonian perturbations of L0L^{0} in which singularities of this type occur.

(c) Repeat (a),(b) for LMCF translators rather than SL mm-folds.

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