A Merging timelike and spacelike surfaces

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How to merge spacelike and timelike surfaces in the case of thermal AdS?
In the following paper, there is a statement which says:

We propose to determine the boundary length by merging smoothly the timelike and spacelike surfaces in such a way that they are homologous to the boundary and have a well-defined first derivative at the merging point.

Can this procedure be done in the case of thermal AdS, where the boundary is at infinity?


[1]: https://arxiv.org/abs/2404.01393
 
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