$\omega$-多次调和函数的强拓扑
The strong topology of $ω$-plurisubharmonic functions
AI总结：
该论文在紧致Kähler流形上证明Monge-Ampère算子在相对有限能量位势与测度集合间为同胚，并推广到全序奇点族，应用于方程解的强稳定性。
摘要：
在$(X,\omega)$紧致Kähler流形上，给定一个模型型包络$\psi\in PSH(X,\omega)$（即一个奇点类型），我们证明Monge-Ampère算子是$\psi$-相对有限能量位势集合与$\psi$-相对能量测度集合之间的同胚，其中这些集合分别配备其强拓扑，该强拓扑定义为弱拓扑的最粗细化，使得相对能量成为连续函数。此外，给定一个全序族$\mathcal{A}$，由具有正总质量且代表不同奇点类型的模型型包络组成，集合$X_{\mathcal{A}}, Y_{\mathcal{A}}$分别定义为当$\psi$在$\overline{\mathcal{A}}$中变化时所有$\psi$-相对有限能量位势的并集和所有$\psi$-相对有限能量测度的并集，它们具有两种自然的强拓扑，这些强拓扑扩展了并集各分量上的强拓扑。我们证明Monge-Ampère算子在$X_{\mathcal{A}}$和$Y_{\mathcal{A}}$之间产生一个同胚。作为应用，我们还证明了当测度具有对$p>1$一致$L^{p}$有界密度且预定奇点全序时，预定复Monge-Ampère方程解序列的强稳定性。
英文摘要：
On $(X,ω)$ compact Kähler manifold, given a model type envelope $ψ\in PSH(X,ω)$ (i.e. a singularity type) we prove that the Monge-Ampère operator is an homeomorphism between the set of $ψ$-relative finite energy potentials and the set of $ψ$-relative energy measures endowed with their strong topologies given as the coarsest refinements of the weak topologies such that the relative energies become continuous. Moreover, given a totally ordered family $\mathcal{A}$ of model type envelopes with positive total mass representing different singularities types, the sets $X_{\mathcal{A}}, Y_{\mathcal{A}}$ given respectively as the union of all $ψ$-relative finite energy potentials and of all $ψ$-relative finite energy measures varying $ψ\in\overline{\mathcal{A}}$ have two natural strong topologies which extends the strong topologies on each component of the unions. We show that the Monge-Ampère operator produces an homeomorphism between $X_{\mathcal{A}}$ and $Y_{\mathcal{A}}$. As an application we also prove the strong stability of a sequence of solutions of prescribed complex Monge-Ampère equations when the measures have uniformly $L^{p}$-bounded densities for $p>1$ and the prescribed singularities are totally ordered.
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