HOLOMORPHONAUTICA · MINI GAME
Plurisubharmonic?
z-plane
z₀ = 0w-plane
w₀ = 0Drag the flat plane to move a point; rotate and pinch/wheel to explore the 3D graph.
Two complex planes, one real-valued function. Move the slices, rotate the graphs, and decide whether the function is psh.
An unknown real-valued function u(z,w) on Δ×Δ is shown through its slices. The left panel always represents z, the right w. Move the parameter in one plane and inspect the 3D graph in the other; then switch roles.
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.
The left panel always represents z and the right always represents w. “Move z” displays w↦u(z₀,w) on the right; “Move w” displays z↦u(z,w₀) on the left. Drag the 3D graph to rotate; use mouse wheel, pinch, or ± to zoom.
For smooth u, plurisubharmonicity is equivalent to positive semidefiniteness of its complex Hessian (Levi matrix). Nonnegative diagonal entries alone do not suffice: a mixed term may create a negative eigenvalue. The oblique complex disc hint lets you explore complex directions.
The coordinate-circle tool lifts a purple circle onto the current 3D graph. The oblique-disc tool chooses a direction (a,b) and draws the restriction as another 3D graph. Numerical circle-mean differences are shown only after answering. These are visual aids rather than mathematical proofs.
After answering, blue/white/red indicate positive/zero/negative curvature in the displayed complex direction. Smooth problems use an analytic Levi form; nonsmooth max/min problems use the finite-radius circle-mean gap as a proxy. Gold curves mark switching loci. Answers are analytically determined, not inferred from numerical samples.
Answer a problem to unlock its formula and explanation. Tap an unlocked card to explore again.
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Drag the flat plane to move a point; rotate and pinch/wheel to explore the 3D graph.