Eigenvalues of elliptic operators and geometric applications

Grigor’yan, Alexander; Netrusov, Yuri; Yau, Shing Tung

doi:10.4310/sdg.2004.v9.n1.a5

Cited by 85 publications

(161 citation statements)

References 45 publications

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“…The following result about the existence of capacitors can be found in [CMa08, Lemma 2.2] (see also [Ko93,GY99,GNY04]). We include a proof for the sake of completeness.…”

Section: Splitting Manifoldsmentioning

confidence: 99%

Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Sabourau

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a consequence of this sweep-out estimate, there exists an embedded, closed (possibly singular) minimal hypersurface whose volume is bounded in terms of the volume of the manifold.

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“…The following result about the existence of capacitors can be found in [CMa08, Lemma 2.2] (see also [Ko93,GY99,GNY04]). We include a proof for the sake of completeness.…”

Section: Splitting Manifoldsmentioning

confidence: 99%

Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Sabourau

2015

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…His result answered a challenge of mine (see [735]) when I met him in Utah in 1989. Grigor'yan, Netrusov and I [276] were able to give a simplified proof and apply the estimate to bound the index of minimal surfaces. There are also works by P. Sarnak (see, e.g., [582,355]) on understanding eigenfunctions for such Riemann surfaces.…”

Section: 3mentioning

confidence: 99%

Perspectives on geometric analysis

Yau¹

2005

Surveys in Differential Geometry

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“…This annular decomposition is then used to estimate Rayleigh quotients, thus bounding the spectrum of the Laplacian of M . Korevaar's method was further developed by Grigor'yan-Yau in [11] and Grigor'yan-Netrusov-Yau in [10] to obtain upper bounds on the eigenvalues of elliptic operators on various metric spaces. In [14] Gromov used a different approach (based on Kato's inequality) to obtain upper bounds for the spectrum of the Laplacian on Kähler manifolds [14].…”

Section: Previous Workmentioning

confidence: 99%

“…In [20] Hassannezhad, combining methods of [6] and [10], obtained upper bounds for eigenvalues of the Laplacian in terms of the conformal invariant MCV (see Definition 1.2) and the volume of the manifold.…”

Section: Previous Workmentioning

confidence: 99%

Width, Ricci curvature, and minimal hypersurfaces

Glynn-Adey¹,

Liokumovich²

2017

J. Differential Geom.

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Abstract. Let (M, g0) be a closed Riemannian manifold of dimension n, for 3 ≤ n ≤ 7, and non-negative Ricci curvature. Let g = φ 2 g0 be a metric in the conformal class of g0. We show that there exists a smooth closed embedded minimal hypersurface in (M, g) of volume bounded by C(n)V n−1 n , where V is the total volume of (M, g). When Ric(M, g0) ≥ −(n − 1) we obtain a similar bound with constant C depending only on n and the volume of (M, g0).Our second result concerns manifolds (M, g) of positive Ricci curvature and dimension at most seven. We obtain an effective version of a theorem of F. C. Marques and A. Neves on the existence of infinitely many minimal hypersurfaces on (M, g) . We show that for any such manifold there exists k minimal hypersurfaces of volume at most CnV sys n−1 (M ), where V denotes the volume of (M, g0) and sys n−1 (M ) is the smallest volume of a non-trivial minimal hypersurface.

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Eigenvalues of elliptic operators and geometric applications

Cited by 85 publications

References 45 publications

Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature

Perspectives on geometric analysis

Width, Ricci curvature, and minimal hypersurfaces

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