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A compactness theorem of the fractional Yamabe problem. Kim, Seunghyeok


Since Schoen raised the question of compactness of the full set of solutions of the Yamabe problem in the $C^0$ topology (in 1988), it had been generally expected that the solution set must be $C^0$-compact unless the underlying manifold is conformally equivalent to the standard sphere. In 2008-09, Khuri, Marques, Schoen himself and Brendle gave the surprising answer that the expectation holds whenever the dimension of the manifold is less than 25 (under the validity of the positive mass theorem whose proof is recently announced by Schoen and Yau) but does not if the dimension is 25 or greater. On the other hand, concerning the fractional Yamabe problem on a conformal infinity of an asymptotically hyperbolic manifold, Kim, Musso, and Wei considered an analogous question and constructed manifolds of high dimensions whose solution sets are $C^0$-noncompact (in 2017). In this talk, we show that the solution set is $C^0$-compact if the conformal infinity is non-umbilic and its dimension is 7 or greater. Our proof provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem. This is joint work with Monica Musso (University of Bath, UK) and Juncheng Wei (University of British Columbia, Canada).

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