Comparison geometry for the Bakry-Emery Ricci tensor

By Guofang Wei, W. Wylie. 2007

677 citations recorded by the source corpus.

For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d3c879daa565ef96f72be45bc833ad031607ad2a

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons (2013, background): “for example, Myers’ theorem, eigenvalues estimates, Li–Yau Harnack inequality, Liouville theorems, and the Cheeger–Gromoll splitting theorem [4–6,14,19,24,26,31].”
Eigenvalues of the drifted Laplacian on complete metric measure spaces (2013, background): “Since the weighted volume ∫ M edv of M is finite ([16], cf [23]), 0 is the least eigenvalue with multiplicity one and nonzero constant functions are the associated eigenfunctions.”
Stability and compactness for complete f -minimal surfaces (2012, methodology): “3 [13] can be applied to obtain the following.”
Remarks on non-compact gradient Ricci solitons (2009, methodology): “rigidity results for Ricci solitons with integrable scalar curvature that should be compared with [11], [12]. Note that, combining Lemma 2.3 in [2] with a volume estimate for weighted manifolds, [8], [21], it follows that the scalar curvature of a shrinking Ricci soliton is p-integrable, for every p”
Stability properties and topology at infinity of $$f$$f-minimal hypersurfaces (2013, result): “the seminal works of Morgan, [29], and Wei and Wylie [46].”
A Liouville-Type Theorem for Smooth Metric Measure Spaces (2010, methodology): “See, for example [4, 7, 9, 12, 8] or [11] and the references therein.”
On Gradient Ricci Solitons with Symmetry (2007, background): “2 in [22] it follows that u is constant (also see [25]).”
Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature (2014, methodology): “If we use the curvature condition Ricf ≥ −K in our geometric proof, then some conditions on f would be required (see [11,37]).”
A warped product version of the Cheeger-Gromoll splitting theorem (2015, neutral): “ub-converges to some v∈ TxM. Let γbe the geodesic with γ(0) = xand γ′(0) = v. Then γis a ray, called an asymptotic ray to γ. Deﬁne ht(y) = t−d(y,γ(t)) +bγ(x), by the standard arguments in for example [WW09], ht is a smooth barrier function to bγ at x. Now we compute ∆f(ht) = −∆f(d(y,γ(t)) ≥ −(n−1) v”
Analysis of weighted Laplacian and applications to Ricci solitons (2011, methodology): “By the Laplace comparison theorem in [28], or cf.”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, background): “, also [5, 6, 20, 21, 38, 51, 57, 64, 77, 78] and the references therein), especially since Perelman’s work on the Poincaré Conjecture [62], and the extension of the Curvature-Dimension condition to the metric-measure space setting by Lott-Sturm-Villani [46, 72].”
Geometry of Complete Gradient Shrinking Ricci Solitons (2009, methodology): “hrinking solitons. To state their result, instead of normalizing the potential function f by (1.6), we will use a diﬀerent normalization by requiring Z M e−f = (4π)n2 . (3.1) Remark 3.1. According to [30, 40] or [7], R M e −f is ﬁnite on a complete gradient shrinking soliton (Mn,gij,f). Hence normal”
Smooth metric measure spaces with non-negative curvature (2011, methodology): “We will refer the readers to [23, 32] for some of the results.”
SECOND-ORDER ESTIMATES AND REGULARITY FOR FULLY NONLINEAR ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS (2012, background): “[37] and references therein), while the gradient Ricci soliton equation takes the form ∇(2)u+ Ric = λg which plays important roles in the theory of Ricci flow and has received extensive study.”