Ricci-Flat Metrics (Analytical Methods)

This page will discuss analytical methods used to solve for Ricci-Flat metrics. Meanwhile, here is the a link to one of my papers where I pursue a somewhat systematic study for Ricci-Flat metrics: Symplectic potentials and resolved Ricci-flat ACG metrics. The authors are Aswin K. Balasubramanian (who was an undergraduate at IITM and currently a graduate student at UT Austin) and Chethan Gowdigere (a post-doc at the Abdus Salam ICTP). The abstract is:


We pursue the symplectic description of toric Kahler manifolds. There exists a general local classification of metrics on toric Kahler manifolds equipped with Hamiltonian two-forms due to Apostolov, Calderbank and Gauduchon(ACG). We derive the symplectic potential for these metrics. Using a method due to Abreu, we relate the symplectic potential to the canonical potential written by Guillemin. This enables us to recover the moment polytope associated with metrics and we thus obtain global information about the metric. We illustrate these general considerations by focusing on six-dimensional Ricci flat metrics and obtain Ricci flat metrics associated with real cones over $L^{pqr}$ and $Y^{pq}$ manifolds. The metrics associated with cones over $Y^{pq}$ manifolds turn out to be partially resolved with two blowup parameters taking special (non-zero)values. For a fixed $Y^{pq}$ manifold, we find explicit metrics for several inequivalent blow-ups parametrised by a natural number k in the range $0<k<p$. We also show that all known examples of resolved metrics such as the resolved conifold and the resolution of $$\mathbb{C}^3/\mathbb{Z}_3$$ also fit the ACG classification.

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