Suppose $(V^{2g}, g, \omega, J)$ is an almost Kahler manifold. ie. $(V,\omega)$ is a symplectic manifold with $\omega$-compatible almost complex structure $J$ ($J$ is a symplectomorphism) and such that $\omega(\cdot, J\cdot)$ coincides with a riemannian metric $g$ on $V$. 


Suppose further that $L$ is a lagrangian submanifold of $V$. Then $L$ has a volume, as defined with respect to the riemannian metric $g$. 


Nonetheless I want to believe that there still is an alternative expression of the metric quantity $vol(L,g)$ which exploits both the (almost) Kahler structure and the fact that the submanifold $L$ is lagrangian (ie. totally isotropic) and that we have a canonical decompostion $TM|L=TL \oplus T JL$, ie. the normal bundle of $L$ is precisely $JL$. 


Can anybody testify or provide some evidence or enlightenment on this matter? 


Alternatively, an even simpler question: if we suppose $V$ is closed, then can we give an expression to the volume of $V$ (wrt g) in terms of $vol(L,g)$ which depends essentially on the circumstances of $L$ being lagrangian and $g=\omega(\cdot, J\cdot)$?



How else can we describe the volume of a lagrangian submanifold in a Kahler manifold?
This is probably closer in spirit to what you're looking for than what you've received in the comments. If $(V^{2m}, J, \omega, g)$ is Calabi-Yau (which for me means that $J$ is integrable, and the first Chern class $c\_1(V) = 0$), then one can say much more. In this case there exists a holomorphic nowhere vanishing $(m,0)$ form $\Omega$, called the "holomorphic volume form." The form $\Omega$ is unique up to multiplication by a nowhere vanishing holomorphic function. We don't need to assume that $\Omega$ is parallel, but Yau's theorem does tell us (if $V$ is compact) that we can change the metric, keeping the Kaehler class $[\omega]$ unchanged, to make $\Omega$ parallel (and consequently also the new metric will be Ricci-flat.) But I have digressed.


In such a situation, if $L$ is Lagrangian, then it is well known that the restriction of $\Omega$ to $L$ is equal to $e^{i \theta} \mathrm{Vol\_L}$, where $\mathrm{Vol}\_L$ is the volume form of $L$ (with the induced metric) and $e^{i \theta}$ is the "phase" of the Lagrangian, where $\theta : L \to \mathbb R/ (2 \pi \mathbb Z)$ is a smooth, multivalued function on $L$. In addition, the mean curvature $H$ of $L$ in $V$ is given by $H = J \nabla \theta$. So the *minimal Lagrangian* submanifolds (vanishing mean curvature) correspond to those with *constant* phase function $\theta = \theta\_0$. In this case, *if $L$ is compact*, then the volume of $L$ is given by


$\mathrm{Vol} (L) = \int\_L \mathrm{Vol}\_L = \int\_L e^{-i \theta\_0} \Omega = e^{- i \theta\_0} [\Omega] \cdot [L],$


which is topological. (It looks complex, but it's actually real, because $[\Omega]$ is a class in $H^g(V, \mathbb C)$.) If you prefer, you can just replace $\Omega$ by $e^{- i \theta\_0} \Omega$ to get rid of the phase factor.


Such "minimal Lagrangian" submanifolds, whose volume is purely topological, are also called *special Lagrangian* submanifolds, and are widely studied in calibrated geometry and differential geometric approaches to mirror symmetry. The best place to start looking is the text "[Riemannian Holonomy Groups and Calibrated Geometry](http://books.google.ca/books?id=3ciqJzDCq7EC&dq=inauthor:%22Dominic+D.+Joyce%22&source=bl&ots=dPtVp-uBKc&sig=hLiZFrkDLbEblYy2R_Qjq6FzPVc&hl=en&sa=X&ei=G10uUNES4u_KAcCogdAL&ved=0CEUQ6AEwAw)" by Dominic Joyce and its multiple references.


I'm not sure how much of this will extend to the case of $J$ non-integrable and $c\_1(V) \neq 0$. I'd have to think about it.


