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We gave the Hardy type identities and inequalities for the divergence type operator Lf,V on smooth metric measure spaces. Additionally, we improved a Rellich type inequality by using the improved Hardy type inequality. Our results improved and included many previously known results as special cases. [ABSTRACT FROM AUTHOR]

We study the geometric structure of weighted Einstein smooth metric measure spaces with weighted harmonic Weyl tensor. A complete local classification is provided, showing that either the underlying manifold is Einstein, or decomposes as a warped product in a specific way. Moreover, if the manifold is complete, then it either is a weighted analogue of a space form, or it belongs to a particular family of Einstein warped products. [ABSTRACT FROM AUTHOR]

In this paper, we prove gradient estimates for positive smooth solutions to the weighted nonlinear p-Laplacian equation Δ p , ϕ u + a u p - 1 log u + λ u p - 1 = 0 on compact smooth metric measure spaces with m-Bakry-Émery Ricci curvature bounded from below, where a, λ and p > 1 are some given constants. We generalize and improve the previous results due to Ma and Zhu [Arch. Math. (Basel), 117 (2021)] on the manifold case since one extends the range of p and does not need to suppose non-negativity on the m-Bakry-Émery Ricci curvature. As applications, we also obtain some Liouville type results for the above equation. [ABSTRACT FROM AUTHOR]

In this paper, we first prove a Kazdan–Warner type identity for the problem of prescribing weighted scalar curvature. We then study weighted Yamabe solitons and prove some rigidity results. [ABSTRACT FROM AUTHOR]

Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space. From the sharp Sobolev inequality, he achieved a breakthrough in the conjecture of isoperimetric inequality for minimal submanifolds. In this paper, we extend Brendle's results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for self-expander. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Li–Yau type gradient estimate for a positive solution to the weighted nonlinear parabolic type equation (Δ ϕ - ∂ t) u (x , t) + a (x , t) u (x , t) ln u (x , t) + b (x , t) u (x , t) = 0 on the complete smooth metric measure space under integral Bakry–Émery Ricci curvature bounds. This estimates optimize the obtained conclusions by Zhang and Zhu (J Funct Anal 275:478–515, 2018). [ABSTRACT FROM AUTHOR]

The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m), the weighted Yamabe problem consists on finding another smooth metric measure space conformal to (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ+μe−ϕ∕m\lambda +\mu {e}^{-\phi /m} for some constants μ\mu and λ\lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m). On the other hand, we prove a version of the conformal Schwarz lemma on (M,g,e−ϕdVg,m)\left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m). All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author's previous results under integral Bakry-Émery Ricci tensor bounds, but also are generalizations of the Wei-Wylie's pointwise results. [ABSTRACT FROM AUTHOR]

In this paper, we establish two gap theorems for ends of smooth metric measure space (M^n, g,e^{-f}dv) with the Bakry-Émery Ricci tensor \operatorname {Ric}_{f}\!\ge -(n-1) in a geodesic ball B_{o}(R) with radius R and center o\in M^n. When \operatorname {Ric}_{f}\ge 0 and f has some degeneration outside B_{o}(R), we show that there exists an \epsilon =\epsilon (n,\sup _{B_{o}(1)}|f|) such that such a space has at most two ends if R\le \epsilon. When \operatorname {Ric}_{f}\ge \frac 12 and f(x)\le \frac 14d^2(x,B_{o}(R))+c for some constant c>0 outside B_{o}(R), we can also get the same gap conclusion. [ABSTRACT FROM AUTHOR]

In this paper, we study the eigenvalue problem of poly-drifting Laplacian on complete smooth metric measure space (M,⟨,⟩,e−ϕdv)\left(M,\langle ,\rangle ,{e}^{-\phi }{\rm{d}}v), with nonnegative weighted Ricci curvature Ricϕ≥0{{\rm{Ric}}}^{\phi }\ge 0 for some ϕ∈C2(M)\phi \in {C}^{2}\left(M), which is uniformly bounded from above, and successfully obtain several universal inequalities of this eigenvalue problem.

The weighted Yamabe flow is the geometric flow introduced to study the weighted Yamabe problem on smooth metric measure spaces. Carlotto, Chodosh and Rubinstein have studied the convergence rate of the Yamabe flow. Inspired by their result, we study in this paper the convergence rate of the weighted Yamabe flow. [ABSTRACT FROM AUTHOR]

In this paper we first discuss weighted mean curvature and volume comparisons on smooth metric measure space (M , g , e - f d v) under the integral Bakry–Émery Ricci tensor bounds. In particular, we add an additional condition on the potential function f to ensure the validity of previous conclusions for some cases proved by the second author. Then, we apply the comparison results to get a new diameter estimate and a fundamental group finiteness under the integral Bakry–Émery Ricci tensor bounds, which sharpens Theorem 1.6 in Wu (J Geom Anal 29:828–867, 2019) and can be viewed as the extension of the works of Myers and Aubry. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove almost-Schur inequalities on closed smooth metric measure spaces, which implies the results of Cheng and De Lellis–Topping whenever the weighted function f is constant.

Let (M n , g , e − ϕ d v) be a smooth metric measure space. In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a nonlinear parabolic partial differential equation (Δ ϕ − ∂ t) u = p u + q u a + 1 in M n × (− ∞ , + ∞) , where p , q ∈ C 2 , 1 (M n × (− ∞ , + ∞)) and a is a positive constant. [ABSTRACT FROM AUTHOR]

Let (M n , g , e − ϕ d v) be a smooth metric measure space. In this paper, we derive a series of gradient estimates and a Harnack inequality for positive solutions of a nonlinear parabolic partial differential equation (Δ ϕ − ∂ t) u = q u + a u (ln u) α in M n × (− ∞ , + ∞) , where q ∈ C 2 , 1 ( M n × (− ∞ , + ∞)) and a , α ∈ R. [ABSTRACT FROM AUTHOR]

We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates, and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry-Émery Ricci tensor. These give generalizations of some work of Petersen-Wei, Aubry, Petersen-Sprouse, Yau and more. [ABSTRACT FROM AUTHOR]

We obtain some De Lellis-Topping type inequalities on the smooth metric measure spaces, some of them are as generalization of De Lellis-Topping type inequality that was proved by X. Cheng [Ann. Global Anal. Geom., 2013, 43: 153-160]. [ABSTRACT FROM AUTHOR]

We introduce a scalar invariant on manifolds with density which is analogous to the renormalized volume coefficient v3 in conformal geometry. We show that this invariant is variational and that shrinking gradient Ricci solitons are stable with respect to the associated W-functional. [ABSTRACT FROM AUTHOR]

We prove that on a smooth metric measure space with m−Bakry-Émery curvature bounded from below by −( m − 1) K for some constant K≥0 (i.e., Ric f, m≥−( m − 1) K), the following degenerate elliptic equation [ABSTRACT FROM AUTHOR]

In this paper, we study elliptic gradient estimates for a nonlinear f -heat equation, which is related to the gradient Ricci soliton and the weighted log-Sobolev constant of smooth metric measure spaces. Precisely, we obtain Hamilton’s and Souplet–Zhang’s gradient estimates for positive solutions to the nonlinear f -heat equation only assuming the ∞ -Bakry–Émery Ricci tensor is bounded below. As applications, we prove parabolic Liouville properties for some kind of ancient solutions to the nonlinear f -heat equation. Some special cases are also discussed. [ABSTRACT FROM AUTHOR]

We first provide an alternative proof of the classical Weitzenböck formula for Einstein four-manifolds using Berger curvature decomposition, motivated by which we establish a unified framework for a Weitzenböck formula for a large class of canonical metrics on four-manifolds. As applications, we classify Einstein four-manifolds and conformally Einstein four-manifolds with half two-nonnegative curvature operator, which in some sense provides a characterization of Kähler-Einstein metrics and Hermitian, Einstein metrics with positive scalar curvature on four-manifolds, respectively. We also discuss the classification of four-dimensional gradient shrinking Ricci solitons with half two-nonnegative curvature operator and half harmonic Weyl curvature. [ABSTRACT FROM AUTHOR]

We propose a natural definition of the weighted σ k -curvature for a manifold with density; i.e. a triple ( M n , g , e − ϕ dvol ) . This definition is intended to capture the key properties of the σ k -curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted σ k -curvature functionals when the weighted σ k -curvature is variational. Second, we characterize the shrinking Gaussians as measures on Euclidean space in terms of the total weighted σ k -curvature functionals. Third, we characterize when the weighted σ k -curvature is variational. These results are all analogues of their conformal counterparts, and in the case k = 1 recover some of the well-known properties of Perelman's W -functional. [ABSTRACT FROM AUTHOR]

Let (M, g, e-f dv) be a complete smooth metric measure space with Bakry-Émery Ricci curvature nonnegative outside a compact set. We prove that the number of ends of such a measure space is finite if f has at most sublinear growth outside the compact set. In particular, we give an explicit upper bound for the number. [ABSTRACT FROM AUTHOR]

New lower bounds of the first nonzero eigenvalue of the weighted p -Laplacian are established on compact smooth metric measure spaces with or without boundaries. Under the assumption of positive lower bound for the m -Bakry–Émery Ricci curvature, the Escobar–Lichnerowicz–Reilly type estimates are proved; under the assumption of nonnegative ∞-Bakry–Émery Ricci curvature and the m -Bakry–Émery Ricci curvature bounded from below by a non-positive constant, the Li–Yau type lower bound estimates are given. The weighted p -Bochner formula and the weighted p -Reilly formula are derived as the key tools for the establishment of the above results. [ABSTRACT FROM AUTHOR]

In this paper we establish gradient estimates for positive solutions to the equation (0.1) △ f u p = − λ u on any smooth metric measure space whose m -Bakry–Émery curvature is bounded from below by − ( m − 1 ) K with K ≥ 0 . These estimates imply Liouville theorems for (0.1) . When p → 1 , our main theorem reduces to the gradient estimate of Wang (2010) [9] . As applications, several Harnack inequalities are obtained. [ABSTRACT FROM AUTHOR]

Motivated and inspired by the recent work of Colding (Acta Math 209(2):229-263, ) and Colding and Minicozzi (Calc Var Partial Differ Equ 49(3-4):1045-1059, ) we derive several families of monotonicity formulas for manifolds with nonnegative Bakry-Emery Ricci curvature, extending the formulas in Colding () and Colding and Minicozzi (). [ABSTRACT FROM AUTHOR]

In this paper, we study complete oriented f -minimal hypersurfaces properly immersed in a cylinder shrinking soliton $$(\mathbb{S}^n \times \mathbb{R},\bar g,f)$$.We prove that such hypersurface with L -index one must be either $$\mathbb{S}^n \times \{ 0\}$$ or $$\mathbb{S}^{n - 1} \times \mathbb{R}$$, where $${S}^{n - 1}$$ denotes the sphere in $$\mathbb{S}^n$$ of the same radius. Also we prove a pinching theorem for them. [ABSTRACT FROM AUTHOR]

We obtain two elliptic gradient estimates for positive solutions to the $$f$$ -heat equation on a complete smooth metric measure space with only Bakry-Émery Ricci tensor bounded below. One is a local sharp Souplet-Zhang's type and the other is a global Hamilton's type. As applications, we prove parabolic Liouville theorems for ancient solutions satisfying some growth restriction near infinity. In particular the Liouville results are suitable for the gradient shrinking or steady Ricci solitons. The estimates of derivation of the $$f$$ -heat kernel are also obtained. [ABSTRACT FROM AUTHOR]

We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's ν-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the κ-noncollapsing property. Finally, we use the energy to prove a precompactness theorem for the space of compact quasi-Einstein smooth metric measure spaces, in the spirit of similar results for Einstein metrics and gradient Ricci solitons. [ABSTRACT FROM AUTHOR]

The porous medium equation arises in different applications to model diffiusive phenomena. In this paper, we obtain several gradient estimates for some porous medium type equations on smooth metric measure space with N-Bakry-Emery Ricci tensor bounded from below. In particular, we improve and generalize some current gradient estimates for the porous medium equations. [ABSTRACT FROM AUTHOR]

Abstract: We obtain some results on the rigidity of complete smooth metric measure spaces with weighted Poincaré inequality, which generalize previous results in Cheng and Zhou (2009), Dung (2012), Dung and Sung (2013), Fu and Li (2010), Fu and Yang (2012), Lam (2010), Li (2005), Li and Wang (2001, 2006), Wang (2012). As an application, we prove that a complete -stable minimal hypersurface in with must have only one end. [Copyright &y& Elsevier]

Abstract: Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor to the setting of smooth metric measure spaces. The obstructions we obtain can be realized as tensorial invariants which are polynomial in the Riemann curvature tensor and its divergence. By taking suitable limits of their tensorial forms, we then find obstructions to the existence of static potentials, generalizing to higher dimensions a result of Bartnik and Tod, and to the existence of potentials for gradient Ricci solitons. [Copyright &y& Elsevier]

Smooth metric measure spaces have been studied from the two different perspectives of Bakry-Emery and Chang-Gursky-Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons and static metrics. In this paper, we describe a natural perspective on smooth metric measure spaces from the point of view of conformai geometry and show how it unites these earlier perspectives within a unified framework. We offer many results and interpretations which illustrate the unifying nature of this perspective, including a natural variational characterization of quasi-Einstein metrics as well as some interesting families of examples of such metrics. [ABSTRACT FROM AUTHOR]

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie. [ABSTRACT FROM AUTHOR]

We consider a smooth metric measure space ( M, g, e dv). Let Δ be its weighted Laplacian. Assuming that λ(Δ) is positive and the m-dimensional Bakry-Émery curvature is bounded below in terms of λ(Δ), we prove a splitting theorem for ( M, g, e dv). This theorem generalizes previous results by Lam and Li-Wang (Trans Am Math Soc 362:5043-5062, ; J Diff Geom 58:501-534, ; see also J Diff Geom 62:143-162, ). [ABSTRACT FROM AUTHOR]

We prove a Yau's type gradient estimate for positive f -harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry–Emery Ricci tensor and the weighted mean curvature are bounded below. As an application, we give a Liouville type result for bounded f -harmonic functions with the Dirichlet boundary condition. Our results do not depend on any assumption on the potential function f. [ABSTRACT FROM AUTHOR]

We establish some new compactness criteria for complete Riemannian manifolds via m -Bakry–Émery and m -modified Ricci curvatures in the spirit of Cheeger et al., (1982). Our results improve and generalize previous compactness theorems for complete Riemannian manifolds via m -Bakry–Émery and m -modified Ricci curvatures obtained by Fernández-López and García-Río (2008), Limoncu (2010, 2012), Qian (1997), Wei and Wylie (2009), and Wu (2018). [ABSTRACT FROM AUTHOR]