A J-holomorphic curve on a punctured disc with finite energy and image contained in a compact target extends smoothly across the puncture when the target has a smooth tame almost complex structure and compatible bounded geometry. Local monotonicity controls escape from small target balls, and the elliptic small-energy estimate followed by elliptic regularity gives the extension. This is the step turning a finite-energy plane obtained by rescaling into a J-holomorphic sphere.

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  1. Pseudoholomorphic curve
  2. Symplectic manifold
  3. Symplectic geometry
  4. Differential geometry
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