Extreme black holes have event horizons which are also zero-temperature, or degenerate, Killing horizons. A certain limiting procedure defines a quasi-Einstein equation which these horizons must satisfy. Solutions are triples (M,g,X) where M is a closed manifold (the horizon), g is a Riemannian quasi-Einstein metric, and X is a 1-form. The case of a closed 1-form X is the so-called static case, which includes near horizon geometries of static extreme black holes. We give a rigidity theorem for quasi-Einstein metrics with M a closed manifold and X a closed 1-form. When the quasi-Einstein constant is positive X must be exact, when it is zero X must vanish, and when it is negative either X vanishes or is not exact and then the manifold is the product of a negative Einstein manifold and a circle. While the cases of this theorem for positive or zero quasi-Einstein constant have been known for some time, the negative case had been incorrectly characterized in the past. In joint work, Eric Bahuaud, Sharmila Gunasekaran, Hari K Kunduri, and I were able to correct the characterization. Very recently Will Wylie was able to improve on our result and complete the rigidity theorem for negative quasi-Einstein constant with closed X.