HOLOMORPHONAUTICA · MINI GAME
Ryokuinzu — Pseudoconvex?
z-plane · dragz = 0
w-plane · Ω_z
Numerical approximation (inaccurate near boundary)
Drag a point inside the disc on the left; observe the corresponding fiber.
Move a point z in one complex plane and watch a domain change in another. Guess whether the hidden domain is pseudoconvex. Inspired by Kiyoshi Oka’s “Ryokuinzu” diagrams.
解答後も観察を続けられます。予想と数学的な理由を見比べてみてください。Keep exploring after answering and compare your prediction with the mathematics.
Answered problems unlock, whether or not your answer was correct.
Drag z inside the unit disc in the left plane; the right plane shows Ω_z = {w : (z,w) ∈ Ω}. The definition of Ω ⊂ Δ × ℂ is hidden until you answer. Study its slices and guess Yes / No: Is Ω pseudoconvex? Each individual planar fiber can be pseudoconvex without the total domain being pseudoconvex.
Oka placed the z- and w-planes side by side and considered how the slices in the w-plane changed as z moved. He called these diagrams “Ryokuinzu” (green-shade diagrams) and is said to have drawn them repeatedly. This game adapts that visualization method to explore pseudoconvexity.
Drag z, enable ghost slices, or scan along circles. For supported problems, hints render either log K_{Ω_z}(w,w) or −log dist(w,∂Ω_z) above the w-plane. Keep w₀ fixed while moving z to study variation. The second observable uses the Hartogs radius R(z,w₀)=dist(w₀,∂Ω_z).
For pseudoconvex Ω, the fiberwise log Bergman kernel is plurisubharmonic, but the converse fails in general. The graph is an exploratory illustration, not a proof. Disc kernels are exact; annulus kernels use a truncated convergent Laurent series. For intersections of discs, the kernel is approximated using finite-degree holomorphic polynomials and is inaccurate especially near boundaries. Hartogs radii are exact for circular fibers. Some transformed slices do not yet support hints and are explicitly marked as unavailable. All correct answers come from analytic criteria.
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Explore the hidden domain Ω ⊂ Δ × ℂ by moving its fibers.
If w₀ leaves the fiber, the value becomes undefined. Drag the 3D graph to rotate it.
Move the fibers; use either of the two hints if you wish.