Examples
In the late 1990s, the Yamabe invariant was computed for large classes of 4-manifolds by Claude LeBrun and his collaborators. In particular, it was shown that most compact complex surfaces have negative, exactly computable Yamabe invariant, and that any Kähler–Einstein metric of negative scalar curvature realizes the Yamabe invariant in dimension 4. It was also shown that the Yamabe invariant of is realized by the Fubini–Study metric, and so is less than that of the 4-sphere. Most of these arguments involve Seiberg–Witten theory, and so are specific to dimension 4.
An important result due to Petean states that if is simply connected and has dimension, then . In light of Perelman's solution of the Poincaré conjecture, it follows that a simply connected -manifold can have negative Yamabe invariant only if . On the other hand, as has already been indicated, simply connected -manifolds do in fact often have negative Yamabe invariants.
Below is a table of some smooth manifolds of dimension three with known Yamabe invariant. Recall is defined above to be .
| notes | ||
|---|---|---|
| the 3-sphere | ||
| the trivial 2-sphere bundle over | ||
| the unique non-orientable 2-sphere bundle over | ||
| computed by Bray and Neves | ||
| computed by Bray and Neves | ||
| the 3-torus |
By an argument due to Anderson, Perelman's results on the Ricci flow imply that the constant-curvature metric on any hyperbolic 3-manifold realizes the Yamabe invariant. This provides us with infinitely many examples of 3-manifolds for which the invariant is both negative and exactly computable.
Read more about this topic: Yamabe Invariant
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)