Essential spectrum of the weighted Laplacian on noncompact manifolds and applications

In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for f -stability index of a constant weighted mean curvature hypersurface with finite weighted volume and isometrically immersed in a shrinking gradient Ricci soliton that admits at least one parallel field globally defined. For such hypersurface, we still give a necessary condition for equality to be achieved in the estimate obtained.where ⊥ symbolizes the projection above the normal bundle of M . The weighted mean curvature vector of M is defined byand its the weighted mean curvature H f is given bywhere H = trA and η is unit outside normal vector field. The hypersurface M is called fminimal when its weighted mean curvature vector H f vanishes identically, and when there exists real constant C such that H f = −Cη, we say the hypersurface M has constant weighted mean curvature.The weighted volume of a measurable set Ω ⊂ M is given byLet B M r be the geodesic ball of M with center in a fixed point o ∈ M and radius r > 0. It is said that the weighted volume of M has polynomial growth if there exists positive numbers α and C such that (1.2) V ol f (B M r ∩ M ) ≤ Cr α f -STABILITY INDEX OF THE CONSTANT WEIGHTED MEAN CURVATURE HYPERSURFACES

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“…where Ric f is the Bakry-Émery Ricci curvature and A is the second fundamental form. For more details, see [10].…”

Section: Introductionmentioning

confidence: 99%

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

Alencar

Rocha

2017

Ann Glob Anal Geom

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“…However, by adapting to the weighted setting a criterion due to R. Brooks and Y. Higuchi [Bro81,Hig01] (cf. [Roc17]), if (2.30) holds then…”

Section: Of [Rs01] Bymentioning

confidence: 99%

Remarks on mean curvature flow solitons in warped products

Colombo¹,

Mari²,

Rigoli³

2020

Discrete &Amp; Continuous Dynamical Systems - S

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We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

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Essential spectrum of the weighted Laplacian on noncompact manifolds and applications

Cited by 2 publications

References 30 publications

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

Remarks on mean curvature flow solitons in warped products

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