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Pseudoholomorphic curve

{c}

For compatible metrics on a Riemann surface and symplectic target,
$$
E(u)=\frac12\int_\Sigma|du|^2
=\int_\Sigma u^*\omega+\int_\Sigma|\bar\partial_Ju|^2.
$$
Thus a J-holomorphic curve minimizes energy among maps in its homology class.


Energy identity for a J-holomorphic curve

Dirichlet energy of a map

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Energy identity for a J-holomorphic curve

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Energy identity for a J-holomorphic curve

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