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The Weyl functional near the Yamabe invariant. (English) Zbl 1048.53020

Les auteurs considèrent sur une variété Riemannienne compact \((M_n, g)\) de dimension \(n\geq 3\) deux fonctionnelles. Celle de Yamabe \(I_g\), son infimum dans la classe conforme \([g]\) de \(g\) est noté \(Y_{[g]}\). On sait que \(Y(M)= \sup Y_{[g]}\) sur toutes les classes conformes de \(M\) est majoré par \(Y(S_n)= n(n-1) \omega^{2/n}_n\). Et la fonctionnelle \(W_{[g]}= \int_M| W_g|^{{n\over 2}}_g dV_g\) où \(| W_g|_g\) est la norme du tenseur de Weyl \(W^i_{jk\ell}\). On note \(W(M)= \inf W_{[g]}\) sur toutes les classes conformes de \(M\). Pour les variétés de \(\dim n\geq 4\) les AA. montrent que pour tout \(\varepsilon> 0\) et tout \(k\) assez grand \((k> \omega(M))\), il existe une classe conforme \(C\) sur \(M\) telle que \(Y_C(M)\geq Y(M)- \varepsilon\) et \(k+ \varepsilon\geq W_C(M)\geq k\). Puis les auteurs étudient certaines variétés Kählériennes compacts de \(\dim n=4\).

MSC:

53C20 Global Riemannian geometry, including pinching
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