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
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
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.
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.