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The Yamabe constant on noncompact manifolds. (English) Zbl 1315.53028

This paper deals with continuity properties of the Yamabe constant of Riemannian manifolds. Namely, given a smooth manifold \(M\), consider the Yamabe map \(Y_M: \mathrm{Metr}(M)\to \mathbb R\cup\{-\infty\}\) that assign to each smooth Riemannian metric \(g\) on \(M\) the Yamabe constant of \((M, g)\). The question is whether \(Y_M\) is continuous, provided that \( \mathrm{Metr}_M\), the set of Riemannian metrics on \(M\), is endowed with some natural topological structure. The authors apply the compact-open \(C^k\)-topology and the fine (also known as strong or Whitney) \(C^k\)-topology on \( \mathrm{Metr}_M\). Besides, a uniform \(C^k\)-topology is introduced on \( \mathrm{Metr}_M\), which turns out to be strictly finer than the compact-open \(C^k\)-topology and strictly coarser than the fine \(C^k\)-topology. The following results are obtained (c.f. Proposition 7.2 in [A. L. Besse, Einstein manifolds. Berlin etc.: Springer-Verlag (1987; Zbl 0613.53001)]).
{ Theorem 1.} Let \(M\) be a nonempty manifold of dimension \(\geq 3\). Then \(Y_M\) is upper semicontinuous with respect to the compact-open \(C^2\)-topology on \( \mathrm{Metr}_M\). If \(M\) is compact or \(Y_M(g)=-\infty\), then \(Y_M\) is continuous at \(g\) with respect to the compact-open \(C^2\)-topology on \( \mathrm{Metr}_M\). If \(M\) is noncompact and \(Y_M(g)> -\infty\), then \(Y_M\) is not continuous at \(g\) for any compact-open \(C^k\)-topology on \( \mathrm{Metr}_M\) with \(k\in \mathbb N\cup\{\infty\}\).
{ Theorem 2.} Let \(M\) be a nonempty manifold of dimension \(\geq 3\). Then \(Y_M\) is upper semicontinuous with respect to the uniform \(C^2\)-topology on \( \mathrm{Metr}_M\). At every metric \(g\in \mathrm{Metr}_M\) which satisfies \(Y_M(g)=-\infty\) or admits constants \(\varepsilon\), \(c\in\mathbb R_{>0}\) with \(| \mathrm{Ric}_g|_g \leq c (1+|\mathrm{scal}_g|)\) and \(|| (\mathrm{scal}_g - \varepsilon)_{-}||_{L^{n/2}(g)}<\infty\), the Yamabe map is continuous with respect to the uniform \(C^2\)-topology on \( \mathrm{Metr}_M\).
Here, \(\mathrm{Ric}_g\) and \(\mathrm{scal}_g\) are respectively the Ricci curvature and the scalar curvature of \(g\), and for \(f\in C^\infty (M,\mathbb R)\) the function \(f_{-}\in C^0(M,\mathbb R_{\geq 0})\) is defined by \(f_{-}(x)=- \min\{0, f(x)\}\).
{ Theorem 3.} Let \(M\) be a nonempty manifold of dimension \(\geq 3\). Then \(Y_M\) is continuous with respect to the fine \(C^2\)-topology on \( \mathrm{Metr}_M\).
So, the authors claim that the fine \(C^2\)-topology is the correct topology on \( \mathrm{Metr}_M\) in the context of the Yamabe map on noncompact manifolds.
Moreover, the authors discuss another functional, the Yamabe constant at infinity, \(\bar Y_M\), which was introduced earlier as an analogue of the Yamabe constant for noncompact Riemannian manifolds [S. Kim, Nonlinear Anal., Theory Methods Appl. 26, No. 12, 1985–1993 (1996; Zbl 0858.53029); Geom. Dedicata 64, No. 3, 373–381 (1997; Zbl 0878.53037)]. The following statement on the continuity of \(\bar Y_M\) is proved.
{ Theorem 4.} Let \(M\) be a noncompact manifold of dimension \(\geq 3\). Then \(\bar Y_M\) is locally constant (in particular, continuous) with respect to the fine \(C^2\)-topology on \( \mathrm{Metr}_M\).
Some other general properties of \(Y_M\) and \(\bar Y_M\) are analyzed too.

MSC:

53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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References:

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