201104.3190v1 Regularity and convergence of 4-dimensional extremal Kahler metrics精品.pdfVIP

1 1 0 2 r p A 6 1 Regularity and convergence of 4-dimensional extremal ] K¨ahler metrics G D . h Brian Weber t a m March 11, 2011 [ 1 v 0 Abstract 9 1 We establish a regularity result for the metric on any 4-dimensional extremal K¨ahler 3. manifold, and a weak compactness theorem on the space of such metrics. Specifi- 2 4 cally, the sectional curvature at a point is bounded when the quantity L (|Rm |) in 0 a surrounding ball is sufficiently small compared to the pointwise norm of its scalar 1 curvature. Consequently sequences of 4-dimensional extremal K¨ahler metrics with uni- 1 formly bounded Calabi energies and scalar curvature have convergent subsequences in 0 the Gromov-Hausdorff topology. Gromov-Hausdorff limits are length spaces with the 2 structure of Riemannian orbifolds away from finitely many point-like singularities of unknown structure. y r o e h 1 Introduction T r e b The metric on a compact K¨ahler manifold (M, J, g) is called extremal if it minimizes the m Calabi functional u C (g ) = R2 dµg N M / m among all K¨ahler metrics with homologous K¨ahler forms [10]. The Euler-Lagrange equations o are R,¯ı¯ = 0 for i, j ∈ {1, . . . , n }; for instance constant scalar curvature