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In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed. [ABSTRACT FROM AUTHOR]

We introduce a Yamabe-type flow$ \left\{ \begin{array}{ll} \frac{\partial g}{\partial t} & = (r^m_{\phi}-R^m_{\phi})g \\ \frac{\partial \phi}{\partial t} & = \frac{m}{2}(R^m_{\phi}-r^m_{\phi}) \end{array} \right. \; \; \mbox{ in }M \; \; \mbox{ and }\; \; H^m_{\phi} = 0 \; \; \mbox{ on }\partial M $on a smooth metric measure space with boundary $ (M,g, v^mdV_g,v^mdA_g,m) $, where $ R^m_{\phi} $ is the associated weighted scalar curvature, $ r^m_{\phi} $ is the average of the weighted scalar curvature, and $ H^m_{\phi} $ is the weighted mean curvature. We prove the long-time existence and convergence of this flow. [ABSTRACT FROM AUTHOR]

In this article, we present new gradient estimates for positive solutions to a class of non‐linear elliptic equations involving the f‐Laplacian on a smooth metric measure space. The gradient estimates of interest are of Souplet–Zhang and Hamilton types, respectively, and are established under natural lower bounds on the generalised Bakry–Émery Ricci curvature tensor. From these estimates, we derive amongst other things Harnack inequalities and general global constancy and Liouville‐type theorems. The results and approach undertaken here provide a unified treatment and extend and improve various existing results in the literature. Some implications and applications are presented and discussed. [ABSTRACT FROM AUTHOR]

This article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a (k , m) -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results. [ABSTRACT FROM AUTHOR]

In this paper, we study vanishing and splitting results on a complete smooth metric measure space (M n , g , e - f d v) with various negative m-Bakry-Émery Ricci curvature lower bounds in terms of the first eigenvalue λ 1 (Δ f) of the weighted Laplacian Δ f , i.e., Ric m , n ≥ - a λ 1 (Δ f) - b for 0 < a ≤ m m - 1 , b ≥ 0 . In particular, we consider three main cases for different a and b with or without conditions on λ 1 (Δ f) . These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung, and Vieira. [ABSTRACT FROM AUTHOR]

In this paper, we study weighted p -harmonic forms on smooth metric measure space (M , g , e − f d v) with a weighted Sobolev or a weighted Poincaré inequality. When f is constant, we derive a splitting theorem for Kähler manifolds with maximal bottom spectrum for the p -Laplacian. For general f , we also obtain various splitting and vanishing theorems when the weighted curvature operator of M is bounded below. As applications, we conclude Liouville property for weighted p -harmonic functions and p -harmonic maps. [ABSTRACT FROM AUTHOR]

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to nonlinear parabolic equations involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman–Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry–Émery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we successfully give two interesting lower bounds for the first eigenvalue of submanifolds (with bounded mean curvature) in a hyperbolic space. More precisely, let M be an n -dimensional complete noncompact submanifold in a hyperbolic space and the norm of its mean curvature vector ‖ H ‖ satisfies ‖ H ‖ ⩽ α < n − 1 , then we prove that the first eigenvalue λ 1 , p ( M ) of the p -Laplacian Δ p on M satisfies λ 1 , p ( M ) ⩾ ( n − 1 − α p ) p , 1 < p < ∞ , with equality achieved when M is totally geodesic and p = 2 ; let ( M , g , e − φ d v g ) be an n -dimensional complete noncompact smooth metric measure space with M being a submanifold in a hyperbolic space, and ‖ H ‖ ⩽ α < n − 1 , ‖ ∇ φ ‖ ⩽ C with ∇ the gradient operator on M , then we show that the first eigenvalue λ 1 , φ ( M ) of the weighted Laplacian Δ φ on M satisfies λ 1 , φ ( M ) ⩾ ( n − 1 − α − C ) 2 4 , with equality attained when M is totally geodesic and φ = constant . [ABSTRACT FROM AUTHOR]

In this paper, we first establish a monotonicity formula for vector bundle-valued f -harmonic p -forms on a smooth metric measure space provided 〈 ∇ f , ∇ r 〉 is less that an explicit constant. As applications, we get some vanishing theorems for L 2 f -harmonic forms on concrete geometric models. In the second part, for a metric measure space with nonnegative ∞ -Bakry–Émery–Ricci curvature and with moderate volume growth, we prove that any bounded f -harmonic 1 -form must be parallel. Moreover, some vanishing theorems under nonnegative m -Bakry–Émery–Ricci curvature assumption are also proved. Finally, we consider smooth metric measure spaces with weighted Poincaré inequality and show some vanishing theorems for L q f -harmonic 1 -forms. [ABSTRACT FROM AUTHOR]

In this note, we show an Hamilton type gradient estimate for the solution of time-dependent of nonlinear diffusion equation on smooth metric measure space ( M , g , e − f d v ) . As the application, we show a dimension-free Harnack inequality and prove a Liouville property for f -harmonic functions. [ABSTRACT FROM AUTHOR]

In this paper, we show sharp estimates for the first eigenvalue λ1,p of the weighted p-Laplacian on smooth metric measure spaces (M, g, e−f dv). When the Bakry-Émery curvature Ricf is bounded from below and the weighted function f is of sublinear growth, we prove some rigidity properties provided that the first eigenvalue λ1,p obtains its optimal value. [ABSTRACT FROM AUTHOR]

In this paper, we establish gradient estimates for the positive bounded solutions to a general type of nonlinear parabolic equation concerning the weighted Laplacian ∂ ∂ t − a (x , t) − Δ f u (x , t) = F (u (x , t)) on a smooth metric measure space with the metric evolving under the (k , ∞) -super Perelman–Ricci flow and the Yamabe flow. Applications of our results include Liouville type results and gradient estimates for some important geometric partial differential equations such as the equations involving gradient Ricci solitons and the Einstein-scalar field Lichnerowicz type equations. Our results generalize and improve many previous works. [ABSTRACT FROM AUTHOR]

Let (Mn, g, e- fdv) be a smooth metric measure space of dimensional n. Suppose that v is a positive weighted p-eigenfunction associated to the eigenvalue λ 1, p on M, namelyefdiv (e- f ⊥ v p- 2⊥ v) =- λ 1, pvp- 1, in the distribution sense. We first give a local gradient estimate for v provided the m-dimensional Bakry-Émery curvature Ricfm bounded from below. Consequently, we show that when Ricfm≥; 0 then v is constant if v is of sublinear growth. At the same time, we prove a Harnack inequality for weighted p-harmonic functions. Moreover, we show global sharp gradient estimates for weighted p-eigenfunctions. Then we use these estimates to study geometric structures at infinity when the first eigenvalue λ 1, p is maximal. Our achievements generalize several results proved earlier by Li-Wang, Munteanu-Wang ( [11,12,17,18]). [ABSTRACT FROM AUTHOR]

We first extend Cheeger-Colding's Almost Splitting Theorem (Ann Math 144:189-237, ) to smooth metric measure spaces. Arguments utilizing this extension show that if a smooth metric measure space has almost nonnegative Bakry-Emery Ricci curvature and a lower bound on volume, then its fundamental group is almost abelian. Second, if the smooth metric measure space has Bakry-Emery Ricci curvature bounded from below then the number of generators of the fundamental group is uniformly bounded. These results are extensions of theorems which hold for Riemannian manifolds with Ricci curvature bounded from below. The first result extends a result of Yun (Proc Amer Math Soc 125:1517-1522, ), while the second extends a result of Kapovitch and Wilking (Structure of fundamental groups of manifolds with Ricci curvature bounded below, ). [ABSTRACT FROM AUTHOR]

Abstract: The aim of this note has two folds. First, we show a gradient estimate of the higher eigenfunctions of the weighted Laplacian on smooth metric measure spaces. In the second part, we consider a gradient steady Ricci soliton and prove that there exists a positive constant depending only on the dimension n of the soliton such that there is no nontrivial harmonic 1-form (hence harmonic function) which is in on such a soliton for any . [Copyright &y& Elsevier]

In this paper we prove that on a complete smooth metric measure space with non-negative Bakry–Émery–Ricci curvature if the space of weighted L2 harmonic one-forms is non-trivial, then the weighted volume of the manifold is finite and the universal cover of the manifold splits isometrically as the product of the real line with a hypersurface. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give some results on the topology of manifolds with ∞-Bakry–Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifolds. Notably, we prove existence and vanishing results which generalize to the weighted setting part of Schoen and Yauʼs theory of harmonic maps. [Copyright &y& Elsevier]