Yamabe's problem is the question of the existence on this manifold of a Riemannian metric with constant scalar curvature .
Named after Yamabe , who published the decision in 1960. In 1968, Trudinger discovered a mistake in the proof. In 1984, Trudinger, Aubin And Schön Published a complete solution [1] .
Variations and generalizations
- The non-compact Yamabé problem is as follows: is there a full conformal metric of constant scalar curvature on a smooth complete non-compact Riemannian manifold ? A counterexample was built by Zhiren in 1988.
- The problem of prescribed scalar curvature
Notes
- ↑ Selim Tawfik The Yamabe Problem
Links
- Lee, John Marshall & Parker, Thomas H. (1987), " The Yamabe problem ", Bulletin of the American Mathematical Society Vol. 17: 37–81, doi : 10.1090 / s0273-0979-1987-15514-5 , < http : //www.ams.org/bull/1987-17-01/S0273-0979-1987-15514-5/ > .
- Trudinger, Neil S. (1968), " Remarks concerning the conformal deformation of Riemannian structures on compact manifolds ", Ann. Scuola Norm. Sup. Pisa (3) Vol. 22: 265–274 , < http://www.numdam.org/item?id=ASNSP_1968_3_22_2_265_0 >
- Yamabe, Hidehiko (1960), " On a deformation of Riemannian structures on compact manifolds ", Osaka Journal of Mathematics T. 12: 21–37, ISSN 0030-6126 , < http://projecteuclid.org/euclid.ojm/1200689814 >
- Schoen, Richard (1984), "Conformal deformation of a Riemannian metric to constant scalar curvature", J. Differential Geom. T. 20: 479–495.
- Aubin, Thierry (1976), "Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire", J. Math. Pures Appl. T. (9) 55: 269–296.
- Zhiren, Jin (1988), " A counterexample to the Yamabe problem for complete noncompact manifolds ", Lect. Notes Math. T. 1306: 93–101., Doi : 10.1007 / BFb0082927 , < https://link.springer.com/chapter/10.1007/BFb0082927 >