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Volume et courbure totale pour les hypersurfaces de l'espace euclidien

Alexandru Oancea (2004)

Annales de l’institut Fourier

Nous étudions des analogues en dimension supérieure de l’inégalité de Burago A ( S ) ≤ R 2 T ( S ) , avec S une surface fermée de classe C 2 immergée dans ℝ 3 , A ( S ) son aire et T ( S ) sa courbure totale. Nous donnons un exemple explicite qui prouve qu’une inégalité analogue de la forme vol ( M ) ≤ C n R n T ( M ) , avec C n > 0 une constante, ne peut être vraie pour une hypersurface fermée M de classe C 2 dans ℝ n + 1 , n ≥ 3 . Nous mettons toutefois en évidence une condition suffisante sur la courbure de Ricci sous laquelle l’inégalité est vérifiée en dimension n = 3 . En dimension...

Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Romain Tessera (2007)

Bulletin de la Société Mathématique de France

Let G be a compactly generated locally compact group and let U be a compact generating set. We prove that if G has polynomial growth, then ( U n ) n ∈ ℕ is a Følner sequence and we give a polynomial estimate of the rate of decay of μ ( U n + 1 ∖ U n ) μ ( U n ) . Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain a L p -pointwise...

Weak Capacity and Modulus Comparability in Ahlfors Regular Metric Spaces

Jeff Lindquist (2016)

Analysis and Geometry in Metric Spaces

Let (Z, d, μ) be a compact, connected, Ahlfors Q-regular metric space with Q > 1. Using a hyperbolic filling of Z,we define the notions of the p-capacity between certain subsets of Z and of theweak covering p-capacity of path families Γ in Z.We show comparability results and quasisymmetric invariance.As an application of our methodswe deduce a result due to Tyson on the geometric quasiconformality of quasisymmetric maps between compact, connected Ahlfors Q-regular metric spaces.

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