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hi,

is the calabi conjencture formulated for compact manifolds with boundary ? or only for those without boundary ? excuse me if the question is too trivial but in my literature it isn't mentioned that. hope for answers.

william

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2 Answers 2

No boundary. If you a version with boundary, you have to pick some additional boundary condition, but the boundary will be a real hypersurface in a compact Kaehler manifold with holomorphic volume form, so it will have an enormous collection of differential invariants that you could pick for a boundary condition. At first glance, the most likely choice would be minimality, but I think many other conditions could be used. I don't know what the correct condition would be for applications to string theory, but I am sure the problem has been considered.

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If you rewrite the equation of the Calabi Conjecture as a scalar complex Monge-Ampère equation, then you can just impose Dirichlet boundary conditions for the unknown function. Geometrically, this means that the resulting Kähler-Einstein metric restricted to the boundary Levi distribution will be conformal to the Levi form of the boundary.

The study of the Dirichlet problem for complex Monge-Ampère equations in domains in Euclidean space is a classical topic, with famous works of Bedford-Taylor, Caffarelli-Kohn-Nirenberg-Spruck and many more. In this general formulation on manifolds, which includes Kähler-Einstein metrics with positive Ricci curvature, the problem has been considered and solved recently by Guedj, Kolev and Yeganefar.

Of course, this is just one of the many possible boundary conditions that you can impose, as Ben says.

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yes, but does this work globally? and what is the idea of patching the solutions together? i read yau's proof of the calabi conjencture (that paper with about 70 pages). he worked there only locally (in local coordinates). but he still proved it globally (on the whole compact manifold)? and what would be different by treating a manifold with boundary ? the computations in yau's proof were also locally. –  william Feb 20 '12 at 7:23
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In the original setting of Yau's theorem, only the "higher order" estimates (from $C^3$ up) are local. All the other arguments are global. They follow for example from the maximum principle (of course with some nontrivial local calculations) or integral estimates (Moser iteration), but they cannot be localized as they are. When the manifold has boundary, you can try to apply the same methods, and you encounter new terms that you have to deal with. In the case of max principle, the maximum could be achieved on the boundary. In the case of integral estimates, there will be boundary contributions –  YangMills Feb 20 '12 at 23:21
    
For a very recent survey of the standard techniques for complex Monge-Ampere on complex manifolds with boundary, see this paper of Boucksom: springerlink.com/content/m5n10gk73v388435 –  YangMills Feb 20 '12 at 23:23
    
what happens if not imposing some boundary conditions ? –  william Feb 21 '12 at 8:36
    
can go something wrong ??? –  william Feb 21 '12 at 8:59
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