[Paper Review] The strong topology of $\omega$-plurisubharmonic functions

Antonio Trusiani|arXiv (Cornell University)|Feb 3, 2020

Geometry and complex manifolds10 references4 citations

TL;DR

This paper establishes a strong topological homeomorphism between the space of ω-plurisubharmonic functions with prescribed singularities (relative to a totally ordered family of model-type envelopes) and the corresponding space of non-pluripolar measures with finite relative energy. It proves that the Monge-Ampère operator is a homeomorphism under strong topologies, which are refinements of weak topologies ensuring continuity of relative energy functionals. The key result is the strong stability of solutions to complex Monge-Ampère equations when measures have uniformly bounded L^p densities for p > 1 and singularities are totally ordered.

ABSTRACT

On $(X,\omega)$ compact K\"ahler manifold, given a model type envelope $\psi\in PSH(X,\omega)$ (i.e. a singularity type) we prove that the Monge-Amp\`ere operator is an homeomorphism between the set of $\psi$-relative finite energy potentials and the set of $\psi$-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 $\psi$-relative finite energy potentials and of all $\psi$-relative finite energy measures varying $\psi\in\overline{\mathcal{A}}$ have two natural strong topologies which extends the strong topologies on each component of the unions. We show that the Monge-Amp\`ere 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-Amp\`ere equations when the measures have uniformly $L^{p}$-bounded densities for $p>1$ and the prescribed singularities are totally ordered.

Motivation & Objective

• To extend the Monge-Ampère operator as a homeomorphism between spaces of ω-plurisubharmonic functions and measures with prescribed singularities.
• To define and study strong topologies on unions of such spaces indexed by a totally ordered family of model-type envelopes.
• To establish strong stability of solutions to complex Monge-Ampère equations under uniform L^p bounds on densities and ordered singularities.
• To generalize the known homeomorphism between finite energy potentials and measures to broader, ordered families of singularities.

Proposed method

• Introduces a strong topology on the space of ψ-relative finite energy potentials as the coarsest refinement of the weak topology that makes the ψ-relative energy functional continuous.
• Defines a corresponding strong topology on the space of ψ-relative finite energy measures via continuity of the dual energy functional E∗_ψ.
• Constructs a complete metric dA on the union XA of E1(X, ω, ψ) over a totally ordered family A ⊂ M+, extending the metric d on each component.
• Uses the Finsler metric |f|1,ϕ = ∫_X |f|MAω(ϕ) and the associated distance d(u,v) = E(u)+E(v)−2E(Pω(u,v)) to define the metric topology.
• Applies the theory of relative energies and the relative Monge-Ampère operator to prove continuity and bijectivity of MAω under these topologies.
• Employs compactness, weak convergence, and convergence in capacity arguments to establish stability under L^p bounded densities.

Experimental results

Research questions

• RQ1 Can the Monge-Ampère operator be extended as a homeomorphism between larger spaces of ω-plurisubharmonic functions and measures beyond individual model-type envelopes?
• RQ2 What is the natural strong topology on the union of spaces of potentials and measures indexed by a totally ordered family of singularities?
• RQ3 How does the strong topology on the union relate to the individual strong topologies on each component space?
• RQ4 Under what conditions does the solution sequence of complex Monge-Ampère equations with prescribed singularities converge strongly?
• RQ5 Is there a strong stability result for solutions when the measures have uniformly L^p-bounded densities for p > 1 and singularities are totally ordered?

Key findings

• The Monge-Ampère operator MAω is a homeomorphism between (X_A,norm, d_A) and (Y_A, strong), where X_A is the union of ψ-relative finite energy potentials and Y_A the union of ψ-relative finite energy measures over a totally ordered family A ⊂ M+.
• The strong topology on X_A is the coarsest refinement of the weak topology such that the ψ-relative energy functional E_ψ becomes continuous, and it coincides with the metric topology induced by d_A.
• The strong topology on Y_A is the coarsest refinement of the weak topology such that the dual energy functional E∗_ψ becomes continuous.
• For a sequence of solutions (uk) to MAω(uk) = fkω^n with fk uniformly bounded in L^p (p > 1) and ψk → ψ in A, the solutions converge strongly to u ∈ X_A with MAω(u) = fω^n.
• Convergence in capacity is implied by the strong convergence, as shown via Theorem 6.3.
• The result generalizes the known strong stability under L^p bounds and extends the homeomorphism to ordered families of singularities, resolving a key question in the stability theory of complex Monge-Ampère equations.