The function in a Weinstein structure: terminology and minimal data

Definitions and mathematical statement

Question 1.14 is expressly terminological: "Stein is to Weinstein as plurisubharmonic is to what?" The source explicitly says this is not a mathematical problem. The supplied aid offers "Lyapunov function" for compatibility with a fixed Liouville field, "Weinstein Morse function" when Morse handle data are part of the structure, and sometimes "potential" when all surrounding data are fixed. It reports no universally imposed single word.

A classical open Weinstein structure includes a symplectic form $\omega$, a complete Liouville vector field $Z$, and a proper bounded-below Morse function $\phi$ for which $Z$ is gradient-like. The Liouville condition is $\lambda=\iota_Z\omega$, ${\,\mathrm{d}}\lambda=\omega$. Strict Lyapunov compatibility includes ${\,\mathrm{d}}\phi(Z)>0$ away from the common critical/zero set; the chosen definition may require an additional quantitative or local gradient-like condition. Generalized-Morse variants allow prescribed birth-death singularities. For a compact domain the boundary/outward-field formulation replaces the open exhaustion/completeness conditions. A proposed name should not suppress which of these data are intended.

The aid supplies the sign-checked Stein dictionary. On a complex manifold $(W,J)$, choose ${\,\mathrm{d}}^c\phi=-{\,\mathrm{d}}\phi\circ J$, assume $\omega={\,\mathrm{d}}{\,\mathrm{d}}^c\phi$ is positive, and set $g(v,w)=\omega(v,Jw)$. Then \[ \lambda={\,\mathrm{d}}^c\phi,\qquad \iota_{\nabla_g\phi}\omega=\lambda, \] so the induced Liouville vector field is the Kähler gradient of $\phi$. Completeness, exhaustion and Morse properties are separate conditions, not consequences of the word "strictly plurisubharmonic" alone.

The aid reports three distinctions, illustrated on $({\mathbb{R}}^2,{\,\mathrm{d}} x\wedge{\,\mathrm{d}} y)$. A fixed function need not determine the Liouville field: $\phi=x^2+y^2$ is strictly increasing away from the origin along each distinct complete Liouville field \[ Z_a=\tfrac12(x\partial_x+y\partial_y)+a(-y\partial_x+x\partial_y),\qquad a\in{\mathbb{R}}. \] An exhausting Morse function need not be Lyapunov for a specified field: for $Z_0$, the function $(x-1)^2+y^2$ has ${\,\mathrm{d}}\phi(Z_0)<0$ at $(1/2,0)$. Conversely an exhausting strict Lyapunov function need not be Morse or generalized Morse: take $\phi=F(x^2+y^2)$, with $F(s)=\int_0^s e^{-1/u}\,{\,\mathrm{d}} u$ and the integrand extended by zero at $u=0$. Its critical point is infinitely flat, while ${\,\mathrm{d}}\phi(Z_0)>0$ elsewhere. These are distinctions among properties, not a new nomenclature theorem.

The trailing sentence introducing Takao Akahori belongs to the next question and supplies no additional task here.

Short English statement

The source asks for terminology for the function in a Weinstein structure and explicitly disclaims an open mathematical problem. The supplied aid distinguishes Lyapunov compatibility, Morse data, exhaustion and Liouville completeness, with a sign-fixed plurisubharmonic dictionary.

Sources

• Question 1.14, 2003 AIM workshop Holomorphic curves in contact geometry: https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. The source itself calls the item a question of terminology; the Takao Akahori transition begins the next item.
• Local registry data/unsolvedmath/problems.json, permanent ID 20002072, code catalogue entry; canonical aim-geometry-notes.json, notes[409]; updated 14 August 2026. The unrecoverable statement-status label is retained despite its verification reproducing the terminological question. The original literature field is empty.
• Attempt 1: context only, problem status at run not a problem, rigor label proved, novelty confidence low; current registry open/open triage. The aid's terminology discussion has no identified full primary-source entries. It is machine generated, not peer reviewed, and requires independent verification before citation.