The structure of $${\phi}$$ -stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Abstract

Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as Pis the scalar curvature of (N n , g). In this paper, under a technical assumption on φ, we prove that φ-stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane C or the cylinder R × S 1 .

Stability properties and topology at infinity of $$f$$ f -minimal hypersurfaces

“…Let Hess f (r) := Hess(r) − 1 m ∇f, ∇r · , · then (20)Hess f (r) ≤ h ′ h { · , · − dr ⊗ dr} + 1 m θ(r) · , · (resp. ≥).Note that tracing(20), we recover corresponding estimates for ∆ f r. These are consistent with comparison results for weighted manifolds with Ric f (∇r, ∇r) bounded from below by −(m − 1)G(r) and f satisfying(19) for some non-decreasing function θ ∈ C 0 ([0, +∞)), see Theorem 3.1 in[34].Proof. Observe, first of all, that Hess(r)(∇r, X) = 0 for all X ∈ T x M and x ∈ D o \ {o}.…”

“…This assertion is supported by the following comparison theorem. (18) and (19) hold with ≥, then λ max ≤ h ′ h (r(x)) + 1 m θ(r(x));…”

“…The research on f -minimal hypersurfaces has just started and it has been already approached by many authors; see e.g. [14], [19], [44], [8], [7], [26], [13], [43]. Much effort has been devoted to the study of the stability properties.…”

Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities

“…Hence an isoperimetric hypersurface is also an f -stable one. Recently, many authors have studied complete f -stable minimal surfaces inside 3-manifolds with non-negative f -Ricci tensor or f -scalar curvature, see [18], [22], [13], [17], [29], [25] and [12]. The f -stability of hyperplanes for Euclidean product densities has been considered in [4], see also [16].…”

Free boundary stable hypersurfaces in manifolds with density and rigidity results

“…In the Riemannian setting (constant density f = 1), these definitions coincide with the classical notions of free boundary minimal and stable hypersurfaces. Recently, many authors have considered complete strongly f -stable hypersurfaces with empty boundary, see [16], [19], [14], [12], [15] and [23], among others. However, not much is known about strongly f -stable hypersurfaces with non-empty boundary, and this has been in fact our main motivation in the present work.…”