Some isoperimetric inequalities and eigenvalue estimates

Croke, Christopher B.

doi:10.24033/asens.1390

Cited by 300 publications

(193 citation statements)

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“…(c) It would be interesting however to know whether 1.1 can also be proved under different circumstances, e. g. if there are no curvature conditions but if a lower bound on the injectivity radius is given instead. Such a possibility seems imaginable from the work of Berger ([2], [3]) and Croke [11].…”

Section: Remarks -(A) In 32 We Find More Precisely 'K^2s (N-1) H +1mentioning

confidence: 99%

A note on the isoperimetric constant

Buser¹

1982

Ann. Sci. École Norm. Sup.

407

357

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Section: Remarks -(A) In 32 We Find More Precisely 'K^2s (N-1) H +1mentioning

confidence: 99%

A note on the isoperimetric constant

Buser¹

1982

Ann. Sci. École Norm. Sup.

407

357

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“…Notice dM < 26N for all M in Tl. Then using the Dirichlet drawer principle and the argument in [11,5.2], we get k(6) < N4. Thus we have Theorem 1 with the explicit bound.…”

Section: Proof Theoremmentioning

confidence: 99%

Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds

Yamaguchi¹

1988

Proc. Amer. Math. Soc.

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ABSTRACT. In this paper, we consider a precompact family of Riemannian manifolds with respect to the Hausdorff distance, and prove the homotopy type finiteness of elements in the family. This is an extension in the homotopy type version of the Cheeger and Weinstein finiteness theorems.The notion of Hausdorff distance between Riemannian manifolds was introduced by Gromov [11], and has played an important role in global Riemannian geometry. In [11], he proved the so-called precompactness theorem:THEOREM (GROMOV). For given m,A,D, the family of compact Riemannian m-manifolds M, whose curvatures Rícm o,nd diameters %d\i satisfy Rícm > -(m -1)A2 and &m < D, is precompact with respect to the Hausdorff distance.It was also proved in [11] that the first Betti numbers of manifolds in the above family are uniformly bounded in terms of m, A, D, and that the convergences with repsect to the Hausdorff distance and the Lipschitz distance coincide on the family of compact m-mainfolds M with bounded sectional curvatures |Äm| < A2, volumes voIm < V and injectivity radii %m > £ for given positive constants A, V, e. From the last result, the Cheeger and Weinstein finiteness theorems [4,20] are derived.For related results, see [6,7,8,14,16].The purpose of the present paper is to prove the homotopy type finiteness for any precompact family of complete manifolds whose contractibility radii are uniformly bounded below by a positive constant. The contractibility radius cm of a complete manifold M is defined as the supremum of r such that every metric ball of radius r contains no critical points of the distance function from the center. For the precise definition, see §1. Thus cm is greater than or equal to the injectivity radius imLet 9H be a precompact family of complete m-manifolds with respect to the Hausdorff distance d#. For given R > 0, we set WIr = {M G 971; cm > R}. In this situation, we shall prove the following THEOREM 1. The set of homotopy types of manifolds in Tin is finite.

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“…124] then gives a bound c(M,r) for r < R. Note also that a lower bound on the injectivity radius implies the above inequality [Cr,Proposition 14]: If r < inj(M) then vol(JB(a:,r)) > c(n) ■ rn. In dimension 2, Theorem A below is a satisfying positive answer since the assumptions do not imply a lower bound on the injectivity radius while any weakening of the assumptions leads to immediate counterexamples.…”

Section: (01) Vol(b(xr))>c(mr)?mentioning

confidence: 99%