We construct two types of Eguchi-Hanson metrics with the negative constant scalar curvature. The type I metrics are Kahler. The type II metrics are ALH whose total energy can be negative. We also construct a one-parameter family of complete metrics of Horowitz-Myers type with the negative constant scalar curvature, and verify a positive energy conjecture of Horowitz-Myers for these metrics. Furthermore, We prove the positive energy conjecture for a class of asymptotically Horowitz-Myers metrics on R2×Tn-2, which generalizes the previous results of Barzegar-Chruściel-Hörzinger-Maliborski-Nguyen. The talk is based on the joint works with J Chen and with Z. Liang.