We consider a closed manifold with positive Yamabe invariant and the space of all Yamabe metrics with unit volume on this manifold. Does there exist a neighbourhood of a zero-scalar-curvature Yamabe metric which contains positive-scalar-curvature Yamabe metrics?
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2 Answers
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I am not sure if the property you want is always true, but surely it works in some special cases with symmetries. For instance, it always works for homogeneous metrics on spheres.
For example, take the sphere $S^{2n+1}$ with a $SU(n+1)$-homogeneous metric $g_t$, obtained by rescaling the vertical directions of the Hopf fibration $S^1\to S^{2n+1}\to \mathbb CP^n$ by $t$. Then $g_t$ are (degenerate) Yamabe metrics with constant scalar curvature $scal(g_t)=2n(2n+2-t^2)$ and it is easy to check that the first eigenvalue $\lambda_1(t)$ of the Laplacian of $g_t$ is always larger than $\frac{1}{2n}scal(g_t)$, see e.g. Sec 5 in this paper. The family $g_t$ is obviously a smooth path in the space of metrics (transverse to the foliation by conformal classes), and when $t=\sqrt{2n+2}$ the metric $g_t$ is scalar flat, but for $t<\sqrt{2n+2}$ it has positive scalar curvature.
answered Sep 1, 2013 at 23:35
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$\begingroup$ Hi Renato, does $g_t$ for $t=\sqrt{2n+2}$ have $\Ric(g) = 0$? $\endgroup$ Commented Sep 2, 2013 at 0:33
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1$\begingroup$ Hi Otis! Thanks for giving the full answer, I forgot we even talked about this result a couple weeks ago :) About $g_t$, it is never Ricci-flat. It is easy to compute the Ricci curvature for it, using formulas in Besse. PS: mathoverflow.net/questions/68094/ricci-flat-metric-on-n-sphere $\endgroup$ Commented Sep 2, 2013 at 17:23
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$\begingroup$ Thank you very much! I have read your paper, and it really helped me to understand the situation. $\endgroup$ Commented Nov 13, 2013 at 3:33
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A result of Fischer--Marsden (Duke "Deformations of the Scalar Curvature" 1975) says that if $(M,g)$ is scalar flat, but not Ricci flat, then the scalar curvature map $R :\{metrics\} \to \{functions\}$ maps some small neighborhood of $g$ onto a small neighborhood of $R_g$. So, the answer to your questions in this case is yes, any small neighborhood of $g$ contains a constant positive scalar curvature metric.
On the other hand, any Ricci flat does not have this property (Arms--Marsden "The absence of Killing felds is necessary for linearization stability of Einstein's equations" Indiana Univ Math 1979). This does not answer your question, however, as it does not rule out the possibility that some small neighborhood of $g$ has some $g'$ with $R_{g'}$ a positive constant. I do not know what the answer is in this case.
I have taken some notes on Fischer--Marsden which you may find useful: http://math.stanford.edu/~ochodosh/LinStabNOTES.pdf.
answered Sep 2, 2013 at 0:32
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$\begingroup$ Thank you very much. By googling I had found your note before you kindly mentiond it. It really clarified my understanding of Fischer-Marsden's paper. $\endgroup$ Commented Nov 13, 2013 at 3:37