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

Abstract

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

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

“…1] to weighted minimal submanifolds properly immersed in certain shrinking gradient Ricci solitons (complete weighted manifolds such that Ric − ∇ 2 h = c g, where ∇ 2 denotes the Hessian, Ric is the Ricci tensor, g is the Riemannian metric and c is a positive constant). As recently shown by Alencar and Rocha [1,Thm. 1], the result still holds for a properly immersed submanifold P where the function H h P , ∇h is bounded from above.…”

Minimal submanifolds in a metric measure space

“…Using Theorem 1.2 in [7], Proposition 1.1 in [1], and Proposition 5 in [5], Vieira and we prove the equivalence of properness of immersion, polynomial volume growth and finite weighted volume for submanifols in a shrinking gradient soliton with weighted mean curvature bounded in norm. More precisely,…”

“…1 2 g and jrf j 2 f has polynomial volume growth, it must have finite weighted volume (see [5,Proposition 5]). On the other hand, in [1], Alencar-Rocha proved that if R a complete submanifold immersed in a complete Riemannian manifold with weighted mean curvature vector bounded in norm, then the finite weighted volume of R implies properness of immersion (see [1, Proposition 1.1]).…”

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

“…In [9], it was proved that if a complete properly immersed submanifold Σ in a complete Riemannian manifold M with Ric + ∇ 2 f ≥ 1 2 g and |∇f | 2 ≤ f has polynomial volume growth, it must have finite weighted volume (Proposition 5 in [9]). On the other hand, in [1], Alencar-Rocha proved that if Σ a complete submanifold immersed in a complete Riemannian manifold with weighted mean curvature vector bounded in norm, then the finite weighted volume of Σ implies properness of immersion (Proposition 1.1 in [1]). Using these two properties and Theorem 1.2, we get Theorem 1.3.…”