Yamabe Solitons, Determinant of the Laplacian and the Uniformization Theorem for Riemann Surfaces

In this letter, we prove that non-trivial compact Yamabe solitons or breathers do not exist. In particular our proof in the two dimensional case depends only on properties of the determimant of the Laplacian and turns out to be independent of the classical uniformization theorem (UT). Using this remarkable fact we are able to explain how an independent proof of the UT for Riemann surfaces can be obtained using the Yamabe–Ricci flow. Luca Fabrizio Di Cerbo was partially supported by a Renaissance Technology Fellowship.