Bulletin of The Brazilian Mathematical Society, 2008
In this work we present some properties satisfied by the second L 2-Riemannian Sobolev best const... more In this work we present some properties satisfied by the second L 2-Riemannian Sobolev best constant along the Ricci flow on compact manifolds of dimensions n ≥ 4. We prove that, along the Ricci flow g(t), the second best constant B 0(2, g(t)) depends continuously on t and blows-up in finite time. In certain cases, the speed of the explosion is, at least, the same one of the curvature operator. We also show that, on manifolds with positive curvature operator or pointwise 1/4-pinched curvature, one of the situations holds: B 0(2, g(t)) converges to an explicit constant or extremal functions there exists for t large.
Journal De Mathematiques Pures Et Appliquees, 2011
ABSTRACT Let (Mn,g), n⩾3, be a smooth closed Riemannian manifold with positive scalar curvature R... more ABSTRACT Let (Mn,g), n⩾3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g), which is a geometric invariant, such that Rg⩽n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition, if and only if Mn is diffeomorphic to the standard sphere Sn.RésuméSoit (Mn,g), n⩾3, une variété riemannienne compacte C∞ de courbure scalaire Rg positive. Il existe une constante positive C=C(M,g), qui est un invariant géométrique, telle que Rg⩽n(n−1)C. Dans cet article, on démontre que Rg=n(n−1)C si et seulement si (Mn,g) est isométrique à la sphère euclidienne Sn(C) à courbure sectionnelle C constante. De plus, il existe une métrique riemannienne g sur Mn telle que l'inégalité suivante soit vérifiée, si et seulement si Mn est difféomorphe à la sphère Sn.
Bulletin of The Brazilian Mathematical Society, 2008
In this work we present some properties satisfied by the second L 2-Riemannian Sobolev best const... more In this work we present some properties satisfied by the second L 2-Riemannian Sobolev best constant along the Ricci flow on compact manifolds of dimensions n ≥ 4. We prove that, along the Ricci flow g(t), the second best constant B 0(2, g(t)) depends continuously on t and blows-up in finite time. In certain cases, the speed of the explosion is, at least, the same one of the curvature operator. We also show that, on manifolds with positive curvature operator or pointwise 1/4-pinched curvature, one of the situations holds: B 0(2, g(t)) converges to an explicit constant or extremal functions there exists for t large.
Journal De Mathematiques Pures Et Appliquees, 2011
ABSTRACT Let (Mn,g), n⩾3, be a smooth closed Riemannian manifold with positive scalar curvature R... more ABSTRACT Let (Mn,g), n⩾3, be a smooth closed Riemannian manifold with positive scalar curvature Rg. There exists a positive constant C=C(M,g), which is a geometric invariant, such that Rg⩽n(n−1)C. In this paper we prove that Rg=n(n−1)C if and only if (Mn,g) is isometric to the Euclidean sphere Sn(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on Mn such that the scalar curvature satisfies the pinched condition, if and only if Mn is diffeomorphic to the standard sphere Sn.RésuméSoit (Mn,g), n⩾3, une variété riemannienne compacte C∞ de courbure scalaire Rg positive. Il existe une constante positive C=C(M,g), qui est un invariant géométrique, telle que Rg⩽n(n−1)C. Dans cet article, on démontre que Rg=n(n−1)C si et seulement si (Mn,g) est isométrique à la sphère euclidienne Sn(C) à courbure sectionnelle C constante. De plus, il existe une métrique riemannienne g sur Mn telle que l'inégalité suivante soit vérifiée, si et seulement si Mn est difféomorphe à la sphère Sn.
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