Numerical Kahler-Einstein metric on the third del Pezzo

This webpage collects some of the numerical data and programs described in the paper "Numerical Kahler-Einstein metric on the third del Pezzo" by C. Doran, M. Headrick, C.P. Herzog, J. Kantor, and T. Wiseman, hep-th/0703057.

In particular, here we have the programs for the simulation of Ricci flow in symplectic coordinates, as well as Mathematica notebooks for manipulating that data. We also have notebooks with the method and results of the constrained optimization algorithm.

The c program used to simulate the Ricci flow can be downloaded here.

The flows for the three initial metrics discussed in the write-up, in the form of binary data files, can be downloaded here: first (200 KB), second (500KB), third (200 KB). You should also download this Mathematica notebook, which reads in the data and allows you to plot and manipulate it.

The numerical approximations to the Kahler-Einstein metric, obtained as the fixed points of these numerical simulations, can be downloaded here at three different resolutions: 25 x 25 (30 KB), 100 x 100 (500 KB), 400 x 400 (7 MB). You should also download this Mathematica notebook, which reads in the data and allows you to plot and manipulate it.

Here is the program for simulating the heat equation on the Einstein metric in order to find eigenfunctions and eigenvalues of the scalar Laplacian, and the Mathematica notebook for reading in its output.

Here is the Mathematica notebook that uses the constrained optimization method to compute polynomial approximations to the Einstein symplectic potential up to 18th order. For those who are interested only in the final result, here is a notebook that contains that 18th-order polynomial.

Finally, here are plots of the sectional curvature of the base of the toric fibration, and the Euler density (figures 7 and 8 of the paper), in their full glory: