calabi conjecture on compact manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:45:00Z http://mathoverflow.net/feeds/question/88705 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88705/calabi-conjecture-on-compact-manifolds calabi conjecture on compact manifolds william 2012-02-17T10:54:46Z 2012-02-17T16:48:50Z <p>hi,</p> <p>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.</p> <p>william</p> http://mathoverflow.net/questions/88705/calabi-conjecture-on-compact-manifolds/88706#88706 Answer by Ben McKay for calabi conjecture on compact manifolds Ben McKay 2012-02-17T12:06:35Z 2012-02-17T12:06:35Z <p>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.</p> http://mathoverflow.net/questions/88705/calabi-conjecture-on-compact-manifolds/88735#88735 Answer by YangMills for calabi conjecture on compact manifolds YangMills 2012-02-17T16:48:50Z 2012-02-17T16:48:50Z <p>If you rewrite the equation of the Calabi Conjecture as a scalar complex Monge-Amp&egrave;re equation, then you can just impose Dirichlet boundary conditions for the unknown function. Geometrically, this means that the resulting K&auml;hler-Einstein metric restricted to the boundary Levi distribution will be conformal to the Levi form of the boundary.</p> <p>The study of the Dirichlet problem for complex Monge-Amp&egrave;re 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 K&auml;hler-Einstein metrics with positive Ricci curvature, the problem has been considered and solved recently by <a href="http://arxiv.org/abs/1111.5320" rel="nofollow">Guedj, Kolev and Yeganefar</a>.</p> <p>Of course, this is just one of the many possible boundary conditions that you can impose, as Ben says.</p>