Levi-Civita Ricci-flat metrics on non-K\"{a}hler Calabi-Yau manifolds

TL;DR

This paper constructs new examples of Levi-Civita Ricci-flat Hermitian metrics on various non-Kähler Calabi-Yau manifolds, expanding the known landscape and exploring applications to the Chern-Ricci flow.

Contribution

It introduces novel Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds, including Weyl-Einstein, hyperKähler, and generalized Hopf manifolds, and studies their geometric flows.

Findings

New Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds

Examples of manifolds with negative scalar curvature

Descriptions of Gromov-Hausdorff limits in Ricci flow

Abstract

In this paper, we provide new examples of Levi-Civita Ricci-flat Hermitian metrics on certain compact non-K\"{a}hler Calabi-Yau manifolds, including every compact Hermitian Weyl-Einstein manifold, every compact locally conformal hyperK\"{a}hler manifold, certain suspensions of Brieskorn manifolds, and every generalized Hopf manifold provided by suspensions of exotic spheres. These examples generalize previous constructions on Hopf manifolds. Additionally, we also construct new examples of compact Hermitian manifolds with nonnegative first Chern class that admit constant strictly negative Riemannian scalar curvature. Further, we remark some applications of our main results in the study of the Chern-Ricci flow on compact Hermitian Weyl-Einstein manifolds. In particular, we describe the Gromov-Hausdorff limit for certain explicit finite-time collapsing solutions which generalize previous constructions on Hopf manifolds.