Universal slow fall-off to the unique AdS infinity in Einstein-Gauss-Bonnet gravity

In this paper, the following two propositions are proven under the dominant energy condition for the matter field in the higher-dimensional spherically symmetric spacetime in Einstein-Gauss-Bonnet gravity in the presence of a cosmological constant $\Lambda$. First, for $\Lambda\le 0$ and $\alpha \ge 0$ without a fine-tuning to give a unique anti-de Sitter vacuum, where $\alpha$ is the Gauss-Bonnet coupling constant, vanishing generalized Misner-Sharp mass is equivalent to the maximally symmetric spacetime. Under the fine-tuning, it is equivalent to the vacuum class I spacetime. Second, under the fine-tuning with $\alpha>0$, the asymptotically anti-de Sitter spacetime in the higher-dimensional Henneaux-Teitelboim sense is only a special class of the vacuum class I spacetime. The latter means the universal slow fall-off to the unique anti-de Sitter infinity in the presence of physically reasonable matter.