Timeline for A corollary of the non-existence of positive scalar curvature

9 events

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May 10, 2021 at 22:06	history	became hot network question
May 10, 2021 at 15:53	vote	accept	L.F. Cavenaghi
May 10, 2021 at 15:29	answer	added	Willie Wong		timeline score: 14
May 10, 2021 at 15:25	history	edited	L.F. Cavenaghi	CC BY-SA 4.0	added 109 characters in body
May 10, 2021 at 15:15	history	edited	Francesco Polizzi	CC BY-SA 4.0	added 15 characters in body
May 10, 2021 at 15:05	comment	added	Francesco Polizzi		@WillieWong: oh right, the volume form varies with $f$. Thanks :)
May 10, 2021 at 14:50	comment	added	Willie Wong		@FrancescoPolizzi: you forgot that there is a weight (coming from the volume form of the metric that realizes that value of scalar curvature). (Not saying that the claim is correct or not, but that it is plausible.)
May 10, 2021 at 14:47	comment	added	Francesco Polizzi		There is something I do not understand. Assume that $\dim M \geq 3$ and that $M$ does not admit a metric of positive scalar curvature. By Kazhdan-Werner trichotomy theorem, a non-zero function $f$ is the scalar curvature of a Riemannian metric on $M$ if and only if it is negative somewhere. So, you are asserting that, as soon as a smooth function $f$ is negative at some point of $M$, then its integral on $M$ is non-positive. How is this possible? Or am I missing something?
May 10, 2021 at 14:02	history	asked	L.F. Cavenaghi	CC BY-SA 4.0