Abstract
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.
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Besse L.A.: Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathamatics and Related Areas (3)], vol. 10. Springer, Berlin (1987)
Chen, X., Peng, L., Tian, G.: A note on uniformization theorem of Riemann surfaces by Ricci flow. Proc. AMS (to appear), arXiv:math.DG/0505163
Chow B. (1991). The Ricci flow on he 2-sphere. J. Differe. Geom. 33(2): 325–334
Chow B. (1991). On the entropy estimate for the Ricci flow on compact 2-orbifolds. J. Differe. Geom. 33(2): 597–600
Chow B., Lu P. and Ni L. (2006). Hamilton’s Ricci Flow. American Mathematical Society/Science Press, Providence
Gilkey P.B. (1995). Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem. CRC Press, Boca Raton
Hamilton, R.S.: The Ricci flow on surfaces, mathematics and general relativity, Santa Cruz, SA, 1986. Contemp. Math, vol. 71, pp. 237–262. American Mathematical Society, Providence (1988)
Perelman, G.A.: The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159
Polyakov A. (1981). Quantum geometry of Fermionic strings. Phys. Lett. B 103: 211–213
Osgood B., Phillips R. and Sarnak P. (1988). Extremals of determinants of Laplacians. J. Funct. Anal. 80: 148–211
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Luca Fabrizio Di Cerbo was partially supported by a Renaissance Technology Fellowship.
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Di Cerbo, L.F., Disconzi, M.M. Yamabe Solitons, Determinant of the Laplacian and the Uniformization Theorem for Riemann Surfaces. Lett Math Phys 83, 13–18 (2008). https://doi.org/10.1007/s11005-007-0195-6
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DOI: https://doi.org/10.1007/s11005-007-0195-6