Geometric inequalities for manifolds with Ricci curvature in the Kato class

Carron, Gilles

Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 3095-3167.

On démontre qu’une variété riemannienne complète vérifiant une inégalité de Sobolev euclidienne et dont la courbure de Ricci est petite dans une classe de Kato et à croissance euclidienne du volume. On obtient aussi des estimations spectrales, du noyau de la chaleur et du premier nombre de Betti des variétés riemanniennes compactes dont la courbure de Ricci est controlée dans une classe de Kato.

We obtain Euclidean volume growth results for complete Riemannian manifolds satisfying a Euclidean Sobolev inequality and a spectral type condition on the Ricci curvature. We also obtain eigenvalue estimates, heat kernel estimates, and Betti number estimates for closed manifolds whose Ricci curvature is controlled in the Kato class.

Classification : 53C21, 58J35, 58C40, 58J50
Keywords: Sobolev inequalities, volume growth, Green kernel, Doob transform
Mot clés : Inégalité de Sobolev, croissance du volume, noyau de Green, transformée de Doob

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