Perron-Bremermann Envelope in Complex Analysis

• The Perron-Bremermann envelope is the supremum over admissible plurisubharmonic subsolutions subject to boundary, obstacle, or cohomological constraints.
• It generalizes the classical Perron envelope to several complex variables, playing a crucial role in solving Dirichlet and obstacle problems on complex domains and compact Kähler manifolds.
• Applications include degenerate Monge–Ampère equations, q-plurisubharmonic Dirichlet theory, and prescribed singularity problems in positive metrics, with strong regularity results like C¹,¹ continuity.

The Perron–Bremermann envelope is the pluripotential-theoretic analogue of the classical Perron envelope for harmonic functions: it is obtained by taking the pointwise supremum of admissible subsolutions, typically plurisubharmonic functions constrained by boundary data, an obstacle, or a global cohomological condition. In several complex variables and complex geometry, this construction appears in Dirichlet problems on bounded domains, in obstacle problems on compact Kähler manifolds, in degenerate complex Monge–Ampère equations, in $q$-plurisubharmonic Dirichlet theory, and in envelopes of positive metrics with prescribed singularities (Tosatti, 2017, Nilsson, 2021, Dieu et al., 15 Sep 2025, Pawlaschyk, 15 Jun 2026, Benelkourchi, 2017, Do et al., 2021, Ross et al., 2012).

1. Classical construction in potential theory and pluripotential theory

In classical potential theory, for a domain $\Omega \subset \mathbb{R}^n$ and continuous boundary data $\varphi$ on $\partial\Omega$, the Perron envelope is defined by

$$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$

Under suitable conditions, this envelope is harmonic and solves the Dirichlet problem.

In complex analysis and pluripotential theory, one replaces “subharmonic” by plurisubharmonic and domains in $\mathbb{R}^n$ by domains in $\mathbb{C}^n$. For a bounded domain $\Omega\subset\mathbb{C}^n$ and a continuous function $\varphi$ on $\partial\Omega$ or $\Omega \subset \mathbb{R}^n$0, a typical envelope is

$\Omega \subset \mathbb{R}^n$1

More specifically, the Perron–Bremermann envelope in several complex variables is a Perron-type envelope built from plurisubharmonic subsolutions: $\Omega \subset \mathbb{R}^n$2 More generally, one can define obstacle envelopes

$\Omega \subset \mathbb{R}^n$3

The common structure is the supremum of a family of subsolutions satisfying a boundary or obstacle constraint, with the envelope serving as the canonical candidate for a Dirichlet-type solution (Tosatti, 2017, Benelkourchi, 2017).

2. Boundary data, singularities, and continuity on bounded domains

On a bounded domain $\Omega \subset \mathbb{R}^n$4, one version used for unbounded or singular data is

$\Omega \subset \mathbb{R}^n$5

Here $\Omega \subset \mathbb{R}^n$6 is the upper semicontinuous regularization on $\Omega \subset \mathbb{R}^n$7. In the setting of bounded $\Omega \subset \mathbb{R}^n$8-regular domains, continuity of the envelope can be recovered for data $\Omega \subset \mathbb{R}^n$9 satisfying $\varphi$0 on $\varphi$1 and appropriate control at singularities via strong majorants and strong minorants. If $\varphi$2 is bounded from below and has a strong majorant $\varphi$3, then $\varphi$4 is continuous on

$\varphi$5

If $\varphi$6 has both a strong minorant $\varphi$7 and a strong majorant $\varphi$8, then $\varphi$9 is continuous on

$\partial\Omega$0

These results are presented as a generalization of Walsh’s classical continuity theorem to envelopes with unbounded and potentially singular boundary data (Nilsson, 2021).

A further extension replaces $\partial\Omega$1-regularity by quasi $\partial\Omega$2-regularity. For a bounded regular domain $\partial\Omega$3, quasi $\partial\Omega$4-regularity means that for every continuous $\partial\Omega$5, there exist an $\partial\Omega$6-pluripolar set $\partial\Omega$7 and a quasi upper bounded $\partial\Omega$8 such that

$\partial\Omega$9

In this setting, the paper studies

$$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$0

and proves that if $$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$1 is nearly continuous and $$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$2 admits a superharmonic majorant $$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$3, then $$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$4 is plurisubharmonic; under additional hypotheses, it is quasi upper bounded, attains the prescribed boundary values outside a pluripolar set, and is continuous at every point of

$$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$5

Under suitable assumptions, $$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$6 is also the unique quasi upper bounded solution of the generalized Dirichlet problem for maximal plurisubharmonic functions (Dieu et al., 15 Sep 2025).

3. Global obstacle envelopes in Kähler classes

On a compact Kähler manifold $$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$7, the Perron–Bremermann paradigm becomes a global obstacle problem. If

$$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$8

is a closed real $$P(\varphi)(x):=\sup\{u(x)\mid u\text{ subharmonic on }\Omega,\;\limsup_{z\to \xi}u(z)\le\varphi(\xi)\;\forall\,\xi\in\partial\Omega\}.$$9-form cohomologous to $\mathbb{R}^n$0, with $\mathbb{R}^n$1, the envelope studied in the Kähler setting is

$\mathbb{R}^n$2

Equivalently, one may write

$\mathbb{R}^n$3

This is the exact Perron-type envelope in a Kähler class: given a smooth obstacle $\mathbb{R}^n$4, take the largest $\mathbb{R}^n$5-psh function that stays below $\mathbb{R}^n$6. The same framework includes the rooftop envelope

$\mathbb{R}^n$7

for $\mathbb{R}^n$8.

The central regularity statement is Theorem 1.1: $\mathbb{R}^n$9. More generally, Theorem 3.1 asserts that $\mathbb{C}^n$0 for any $\mathbb{C}^n$1 obstacle functions $\mathbb{C}^n$2. The proof uses Berman’s approximation by Monge–Ampère equations

$\mathbb{C}^n$3

together with the Chu–Tosatti–Weinkove $\mathbb{C}^n$4 estimate machinery to obtain uniform second-derivative bounds

$\mathbb{C}^n$5

The paper states that this resolves affirmatively a conjecture of Berman and that the regularity is “in general optimal” (Tosatti, 2017).

Conceptually, $\mathbb{C}^n$6 plays the same role as the classical Perron–Bremermann envelope: both are defined as a supremum over admissible plurisubharmonic functions satisfying a constraint, but on a compact Kähler manifold the constraint is global, the background form $\mathbb{C}^n$7 encodes cohomological data, and the contact set $\mathbb{C}^n$8 plays the role of a free boundary (Tosatti, 2017).

4. Global Perron envelopes for degenerate complex Monge–Ampère equations

A different global realization appears for degenerate complex Monge–Ampère equations on compact Kähler manifolds. Let $\mathbb{C}^n$9 be compact Kähler, let $\Omega\subset\mathbb{C}^n$0 be a smooth semi-positive representative of a big cohomology class, and consider

$\Omega\subset\mathbb{C}^n$1

where $\Omega\subset\mathbb{C}^n$2 is a non-negative Radon measure on $\Omega\subset\mathbb{C}^n$3 and $\Omega\subset\mathbb{C}^n$4 is measurable. The relevant class is

$\Omega\subset\mathbb{C}^n$5

defined using the non-pluripolar Monge–Ampère product

$\Omega\subset\mathbb{C}^n$6

The subsolution family is

$\Omega\subset\mathbb{C}^n$7

and the solution is constructed as the Perron envelope

$\Omega\subset\mathbb{C}^n$8

Under the hypotheses that $\Omega\subset\mathbb{C}^n$9 is continuous and non-decreasing for every $\varphi$0, that $\varphi$1 for all $\varphi$2, and that

$\varphi$3

the paper proves existence of a solution $\varphi$4 and uniqueness up to additive constant.

The technical basis of the construction is characteristic of Perron theory. The class $\varphi$5 is shown to be nonempty; a Demailly-type inequality is established,

$\varphi$6

so maxima of subsolutions remain subsolutions; Choquet’s lemma furnishes an increasing sequence whose upper semicontinuous regularization is the envelope; and comparison principles imply uniqueness. This is explicitly described as a global Perron–Bremermann type envelope for a degenerate complex Monge–Ampère equation on a compact manifold (Benelkourchi, 2017).

5. $\varphi$7-plurisubharmonic and viscosity generalizations

The envelope method also extends beyond ordinary plurisubharmonicity. For $\varphi$8-plurisubharmonic functions on an unbounded domain $\varphi$9, with $\partial\Omega$0 and $\partial\Omega$1, the $\partial\Omega$2-Perron–Bremermann envelope is defined by

$\partial\Omega$3

If $\partial\Omega$4 has $\partial\Omega$5-peak points at all boundary points and is of bounded type, and if $\partial\Omega$6, then bounded continuous boundary data extend to a maximal bounded continuous function on $\partial\Omega$7 that is $\partial\Omega$8-plurisubharmonic and $\partial\Omega$9-plurisuperharmonic on $\Omega \subset \mathbb{R}^n$00, coincides with the Perron–Bremermann envelope, and is the unique $\Omega \subset \mathbb{R}^n$01-Bremermann function with the prescribed boundary values. For $\Omega \subset \mathbb{R}^n$02-smooth functions, the paper states that the combined $\Omega \subset \mathbb{R}^n$03-plurisubharmonic and $\Omega \subset \mathbb{R}^n$04-plurisuperharmonic condition is linked to

$\Omega \subset \mathbb{R}^n$05

The bounded type hypothesis supplies the global maximum principle needed on unbounded domains (Pawlaschyk, 15 Jun 2026).

A viscosity analogue appears for fully nonlinear elliptic equations of the form

$\Omega \subset \mathbb{R}^n$06

on bounded domains $\Omega \subset \mathbb{R}^n$07, where $\Omega \subset \mathbb{R}^n$08 depends on the eigenvalues of the complex Hessian. Given a bounded viscosity supersolution $\Omega \subset \mathbb{R}^n$09, the Perron–Bremermann envelope is defined by

$\Omega \subset \mathbb{R}^n$10

Under the paper’s comparison hypotheses, $\Omega \subset \mathbb{R}^n$11 is a viscosity subsolution, $\Omega \subset \mathbb{R}^n$12 is a supersolution, and the envelope is a discontinuous viscosity solution. When $\Omega \subset \mathbb{R}^n$13 is independent of $\Omega \subset \mathbb{R}^n$14, Proposition 5.1 states that $\Omega \subset \mathbb{R}^n$15 is a maximal viscosity subsolution, and Theorem 5.2 shows that on each relatively compact open subset $\Omega \subset \mathbb{R}^n$16, such a maximal viscosity subsolution is the decreasing limit of viscosity solutions of the same equation. This yields the approximation statement that, under suitable conditions, a Perron–Bremermann envelope can be approximated by a decreasing sequence of viscosity solutions (Do et al., 2021).

6. Prescribed singularities, equilibrium sets, and metric geometry

For positive metrics on a line bundle $\Omega \subset \mathbb{R}^n$17, the envelope can incorporate a prescribed singularity type. With $\Omega \subset \mathbb{R}^n$18 a reference metric on $\Omega \subset \mathbb{R}^n$19, $\Omega \subset \mathbb{R}^n$20 a positive singular metric on an auxiliary line bundle $\Omega \subset \mathbb{R}^n$21, and

$\Omega \subset \mathbb{R}^n$22

the cutoff envelope is

$\Omega \subset \mathbb{R}^n$23

and the maximal envelope with prescribed singularity type is

$\Omega \subset \mathbb{R}^n$24

with upper semicontinuous regularization

$\Omega \subset \mathbb{R}^n$25

Informally, $\Omega \subset \mathbb{R}^n$26 is the largest positive metric on $\Omega \subset \mathbb{R}^n$27 that lies below $\Omega \subset \mathbb{R}^n$28 and has singularities no worse than those encoded by $\Omega \subset \mathbb{R}^n$29.

The regularity theory parallels the Kähler obstacle problem. If $\Omega \subset \mathbb{R}^n$30 is Lipschitz, respectively $\Omega \subset \mathbb{R}^n$31, then $\Omega \subset \mathbb{R}^n$32 is Lipschitz, respectively $\Omega \subset \mathbb{R}^n$33, on

$\Omega \subset \mathbb{R}^n$34

The Monge–Ampère measure is described through the equilibrium set

$\Omega \subset \mathbb{R}^n$35

and Theorem 1.2 gives

$\Omega \subset \mathbb{R}^n$36

Thus the Monge–Ampère measure of the envelope is supported on the contact set with the obstacle.

The same envelope admits analytic approximations through partial Bergman kernels. On compact subsets of $\Omega \subset \mathbb{R}^n$37,

$\Omega \subset \mathbb{R}^n$38

uniformly, and

$\Omega \subset \mathbb{R}^n$39

The paper also derives a product formula for envelopes with rescaled singularities, identifies the Legendre transform of a test curve of singularity types with a maximal envelope on $\Omega \subset \mathbb{R}^n$40, and connects the associated exhaustion function to the geometry of the Okounkov body. In special cases, the construction reduces to the pluricomplex Green function with prescribed singularities, making explicit its Perron–Bremermann character (Ross et al., 2012).

A coherent synthesis emerges across these settings. The Perron–Bremermann envelope is always a supremum over admissible subsolutions, but the admissible class changes with the problem: $\Omega \subset \mathbb{R}^n$41 on bounded domains, $\Omega \subset \mathbb{R}^n$42 on compact Kähler manifolds, $\Omega \subset \mathbb{R}^n$43 for global Monge–Ampère equations, $\Omega \subset \mathbb{R}^n$44 for $\Omega \subset \mathbb{R}^n$45-plurisubharmonic Dirichlet theory, viscosity subsolutions for fully nonlinear complex Hessian equations, and positive metrics with fixed singularity type in complex geometry. What remains invariant is the role of the envelope as the canonical maximal object under the prescribed constraints, together with its tight relation to maximality, contact sets, and degenerate complex Monge–Ampère structure (Tosatti, 2017, Benelkourchi, 2017, Pawlaschyk, 15 Jun 2026, Do et al., 2021, Ross et al., 2012).