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I am a beginner in symplectic geometry. The questions I ask may be so trivial.

In McDuff's "What is Symplectic Geometry?", she writes:

We saw earlier that the symplectic area of a surface is invariant under deformations of the surface that fix its boundary. (Cf. Figure 1.3 (I).) It follows easily that their metric area can only increase under such deformations, i.e. J-holomorphic curves are so-called $g_J$-minimal surfaces.

What I don't understand is why the symplectic area is invariant under the deformation of with boundary it follows that the metric area can only increase under this deformation.

Any comment is appreciated.

J _ZWei
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  • This is because by definition, a symplectic form $\omega$ is closed so when you integrate the form, you apply Stokes and you are left with an integral that depends on the boundary. – Alvin Jin Aug 05 '22 at 16:41
  • @AlvinJin Thanks for your comment. I know $omega$ is closed induces that symplectic area only depends on the boundary. But I don't why this fact can explain metric area can only increase under such deformations, where $g_J(w,v)=\omega (w,Jv)$. – J _ZWei Aug 05 '22 at 18:38
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    The Wirtinger inequality is the missing important fact. I suspect it's in McDuff's discussion. – Ted Shifrin Aug 06 '22 at 05:07
  • @TedShifrin Thanks! – J _ZWei Aug 06 '22 at 17:32

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