A generalization of Aubin’s result for a Yamabe-type problem on smooth metric measure spaces. (English) Zbl 1479.53052
Summary: The Yamabe problem in compact closed Riemannian manifolds is concerned with finding a metric with constant scalar curvature in the conformal class of a given metric. This problem was solved by the combined work of Yamabe, Trudinger, Aubin, and Schoen. In particular, Aubin solved the case when the Riemannian manifold is compact, non-locally conformally flat and with dimension equal and greater than 6. In [J. Differ. Geom. 101, No. 3, 467–505 (2015; Zbl 1334.53031)], J. S. Case considered a Yamabe-type problem in the setting of smooth measure space in manifolds and for a parameter \(m\), which generalize the original Yamabe problem when \(m = 0\). Also, Case solved this problem when the parameter \(m\) is a natural number. In the context of Yamabe-type problem we generalize Aubin’s result for non-locally conformally flat manifolds, with dimension equal and greater than 6 and small parameter \(m\).
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 53C18 | Conformal structures on manifolds |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
Citations:
Zbl 1334.53031References:
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