Some New Results on Eigenvectors via Dimension, Diameter, and Ricci Curvature☆

By D. Bakry, Zhongmin Qian. 2000

176 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d974d650a351fc9bc123678dff30001003122c23

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue (2012, background): “les of Riemannian manifolds with given diameter bounds and Ricci curvature lower bounds such that the ﬁrst eigenvalue is as close as desired to µ. The construction is similar to that given in [7] and [5], but we include it here because the construction also produces examples proving that the modulus”
Comparison geometry for the Bakry-Emery Ricci tensor (2007, background): “rison theorems for RicN f . The Bochner formula for RicN f has many other applications to other geometric problems not treated here such as eigenvalue problems and Liouville theorems, see for example [5] and [18] and the references there in. In [22] Lott shows that if Mis compact with Ricq f ≥ λfor ”
Sharp estimate on the first eigenvalue of the p-Laplacian (2011, methodology): “ in [9] and [23]. A more sophisticated estimate on |Hess|2 which works in the linear case is 8 DANIELE VALTORTA the following: |Hess|2 ≥ (∆u)2 n + n n−1 ∆u n −A 2 This estimate plays a key role in [1] to get the comparison with the one dimensional model, in particular here is the only point where t”
An optimal anisotropic Poincaré inequality for convex domains (2011, methodology): “ems hard to apply. Instead, we adopt the technique based on gradient comparison with a one dimensional model function, which was developed by Kro¨ger [11] and improved by Chen-Wang [6] and Bakry-Qian [5]. Surprisingly, we ﬁnd that the one dimensional model coincides with that for the Laplacian case.”
Volume comparison theorems without Jacobi fields (2005, background): “— We shall not give the details (see [2] for a complete proof).”
On the geometry of metric measure spaces. II (2006, background): “Also note that according to [3], the Examples (i)–(iii) equipped with natural weighted Laplacians are also the prototypes for the Bakry–Émery curvaturedimension condition.”
Equivalent semigroup properties for the curvature-dimension condition (2010, methodology): “ 1}. The following lower bound of λ 1 is a simple consequence of Theorem 1.1 (2). This estimate is well known as the Lichnerowicz estimate [12] for Z = 0, and was extended to Z 6= 0 by Bakry and Qian [5]. Corollary1.5([12, 5]).Let Z = ∇V such that µ(dx) := eV (x)dx is a probability measure. If (1.1)”
Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition (2011, background): “aces in Euclidean space), so the Riemannian setting is really the right one for studying the CDD condition. One may also try to employ the semi-group approach pioneered by Bakry and Emery´ [3], as in [4, 6], but this approach crucially depends on the existence of good functional versions of the corr”
Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature (2014, background): “Li-Yau estimate has been improved and generalized to other nonlinear equations on a Riemannian manifold, see [1, 2, 5, 7, 9, 15, 16, 19, 20, 21, 22, 24, 25, 29, 30, 31, 34, 35, 36, 41, 42] and references therein.”
Sharp estimates on the first eigenvalue of the $$p$$p-Laplacian with negative Ricci lower bound (2012, methodology): “Qian in [1] to obtain eigenvalue estimates for weighted linear Laplace operators with assumptions on the Bakry–Emery Ricci curvature.”
On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons (2013, methodology): “See [2–6,14,19,20,24,26,31] and references therein.”
Obata’s Rigidity Theorem for Metric Measure Spaces (2014, methodology): “Again, wewill apply ideas of Bakry/Qian [5].”
Moduli of continuity, isoperimetric profiles, and multi-point estimates in geometric heat equations (2014, methodology): “The result of Corollary 7 is closely related to results proved for eigenfunctions using maximum principle estimates applied to the gradient, as in [25,60].”
Spectral gap of positive operators and applications (2000, background): “Qian [ 5 ], M. Saloff-Coste [25] for a lot of literature on the quantitative estimation of spectral gap.”