We study adiabatic limits of Ricci-flat K¨ahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Amp`ere equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metric has Ricci curvature equal to a WeilPetersson metric that measures the variation of complex structure of the Calabi-Yau fibers. This generalizes results of Gross-Wilson for K3 surfaces to higher dimensions.

No abstract uploaded!

No abstract uploaded!

No abstract uploaded!

We introduce a new troubled-cell indicator for the discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. This indicator can be defined on unstructured meshes for high order DG methods and depends only on data from the target cell and its immediate neighbors. It is able to identify shocks without PDE sensitive parameters to tune. Extensive one- and two-dimensional simulations on the hyperbolic systems of Euler equations indicate the good performance of this new troubled-cell indicator coupled with a simple minmod-type TVD limiter for the Runge-Kutta DG (RKDG) methods.