Plurisubharmonic function¶

An upper-semicontinuous function on a complex domain whose restriction to every complex line is subharmonic.

Version: v1 · 2026-09-08 · History
Domain-specific #: 6097
Origin domain: complex analysis
Subdomain: complex analysis
Aliases: PSH function

Core Idea¶

Smooth definitions through a positive semidefinite complex Hessian are equivalent only under regularity, the value minus infinity is allowed and weak-current formulations require conventions. Testing along every holomorphic complex line imposes nonnegative mean curvature in complex directions, yielding functions stable under maxima, limits and constructions of pseudoconvex domains. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of complex analysis. It is the domain-specific identity fixed by the complex domain or analytic space, extended-real function, upper semicontinuity, every complex-line restriction and submean inequality, smooth Levi-form criterion when applicable, weak current formulation and closure and singularity behavior are explicit.

Scope of Application¶

Plurisubharmonic function belongs to complex analysis and is useful where the analyst can specify the typed complex analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the complex domain or analytic space, extended-real function, upper semicontinuity, every complex-line restriction and submean inequality, smooth Levi-form criterion when applicable, weak current formulation and closure and singularity behavior are explicit. The scope is broad within that domain but bounded by the need for the complex domain or analytic space, extended-real function, upper semicontinuity, every complex-line restriction and submean inequality, smooth Levi-form criterion when applicable, weak current formulation and closure and singularity behavior are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the complex domain or analytic space, extended-real function, upper semicontinuity, every complex-line restriction and submean inequality, smooth Levi-form criterion when applicable, weak current formulation and closure and singularity behavior are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Plurisubharmonic function. Plurisubharmonic function compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed complex analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the complex domain or analytic space, extended-real function, upper semicontinuity, every complex-line restriction and submean inequality, smooth Levi-form criterion when applicable, weak current formulation and closure and singularity behavior are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of complex analysis because they reuse the typed complex analysis carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Testing along every holomorphic complex line imposes nonnegative mean curvature in complex directions, yielding functions stable under maxima, limits and constructions of pseudoconvex domains., and type the carrier, state every parameter and convention in the definition, test that the complex domain or analytic space, extended-real function, upper semicontinuity, every complex-line restriction and submean inequality, smooth Levi-form criterion when applicable, weak current formulation and closure and singularity behavior are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Plurisubharmonic function Domain-specific

Parents (1) — more general patterns this builds on

Plurisubharmonic function is a kind of Continuity Prime

The proposed strict upward parent is prime:continuity.

Hierarchy paths (2) — routes to 2 parentless roots

Plurisubharmonic function → Continuity → Neighborhood → Topology

Show alternative path (1)

Neighborhood in Abstraction Space¶

Plurisubharmonic function sits in a crowded region of the domain-specific corpus (13^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Complex Analysis & Integral Transforms (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08