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Dec 5, 2021 at 5:36 comment added Quarto Bendir There is a small error in your post, the given solution is not on R^2, it is only defined on the unit disk or a subset thereof. It also doesn't make sense to take t to infinity (what is the underlying space?). Usually steady solitons are defined for all time, it is not the case here since the metric is not complete. Any complete steady soliton has nonnegative scalar curvature. (theorem of Bing-Long Chen)
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