Topology of three-manifolds with positive 𝑃-scalar curvature

Abstract

Abstract. Consider an n-dimensional smooth Riemannian manifold (M n , g) with a given smooth measure m on it. We call such a triple (M n , g, m) a Riemannian measure space. Perelman introduced a variant of scalar curvature in his recent work on solving Poincaré's conjectureg , where dm = φ dvol(g) and R is the scalar curvature of (M n , g). In this note, we study the topological obstruction for the φ-stable minimal submanifold with positive P -scalar curvature in dimension three under the setting of manifolds with density. Manifold with densityBy a Riemannian measure space we mean a triple (M n , g, m), where (M n , g) is an n-dimensional smooth oriented Riemannian manifold and m is a smooth measure defined on M n . Given a Riemannian manifold, there is a natural measure associated with it, i.e. the Riemannian volume measure dvol(g). By the Radon-Nikodým theorem, there exists a smooth function φ > 0 on M n such that dm = φ dvol(g).Here φ > 0 is called the density of the manifold. The triple (M n , g, φ) is called the Riemannian manifold with density φ. Clearly, the study of Riemannian measure space is equivalent to the study of manifolds with density.The study of manifolds with density traces back to the work of Bakry-Émery [1] in the early 1980s, in which they introduced the Bakry-Émery-Ricci tensor in the study of the diffusion process:

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

“…A two-sided hypersurface P in a model manifold with weight e h is strongly h-stable if it has constant weighted mean curvature H h P , and it is a second order minimum of the functional A h − H h P V h under compactly supported variations (here A h and V h stand for the weighted area and volume functionals). The study of these hypersurfaces has been focus of attention in the last years, with special emphasis in the minimal case, see for instance Fan [27], Ho [39], Colding and Minicozzi [18,19], Liu [45], Cheng, Mejia and Zhou [14], Impera and Rimoldi [44], and Espinar [25]. The h-parabolicity condition for a two-sided hypersurface P entails the existence of a sequence of smooth functions with compact support on P approximating the constant function 1, see Theorem 2.2.…”

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

“…The φ-stable minimal hypersurface in manifold with positive P-scalar curvature has been studied by Fan [3]. In particular, he proved that (Theorem 2.1 in [3]) there is no compact immersed φ-stable minimal two-dimensional submanifold with positive genus in (N 3 , g, φ) where R(g) m ∞ > 0.…”

“…In particular, he proved that (Theorem 2.1 in [3]) there is no compact immersed φ-stable minimal two-dimensional submanifold with positive genus in (N 3 , g, φ) where R(g) m ∞ > 0. Moreover, he proved that (Theorem 2.8 in [3]) if M n−1 is a complete non-compact φ-stable minimal hypersurface in (N n , g, φ) with Rc m ∞ (g) ≥ λ for some positive constant λ, then M is totally geodesic if and only if V ol φ (M, g| M ) = M φ dvol(g| M ) < ∞ (See also [7]).…”

Free boundary stable hypersurfaces in manifolds with density and rigidity results

“…In particular, if Ric f > 0 or II > 0, then there are no compact strongly f -stable hypersurfaces in M . This property was observed by Simons [38] for the Riemannian case, and later generalized by Fan [16], and Cheng, Mejia and Zhou [12] for hypersurfaces with empty boundary in manifolds with density. On the other hand, Espinar showed in [15] that Lemma 4.1 also holds for complete strongly f -stable hypersurfaces of finite type and empty boundary.…”

“…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.…”