Nonnegative Bakry Research Articles

Graphs with nonnegative curvature outside a finite subset, harmonic functions, and number of ends

AbstractWe study graphs with nonnegative Bakry–Émery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov–Hausdorff convergence, we prove that the space of bounded harmonic functions is finite dimensional and, as a corollary, the number of nonparabolic ends is finite.

Sobolev inequality on manifolds with asymptotically nonnegative Bakry–Émery Ricci curvature

AbstractIn this paper, inspired by Brendle (Comm. Pure Appl. Math. 76 (2023), 2192) and Johne (arXiv:2103.08496, 2021), we prove a Sobolev inequality on manifolds with density and asymptotically nonnegative Bakry–Émery Ricci curvature.

Liouville theorems for ancient solutions to the [formula omitted]-harmonic map heat flows

We investigate the Liouville properties of ancient solutions to the V-harmonic map heat flows from complete noncompact Riemannian manifolds with nonnegative Bakry–Emery Ricci curvature. When the target is a simply connected complete Riemannian manifold with nonpositive sectional curvature, then a Liouville theorem of ancient solutions holds under certain growth condition near infinity. When the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, we show that if the image of the ancient solution u is contained in a regular ball near infinity, then u is a constant.

A Schwarz lemma and a Liouville theorem for generalized harmonic maps

When the sectional curvature of the target manifold is negative, we establish a Schwarz lemma for f-harmonic maps, if the dimension of the domain and the target is large, the result improves Theorem 3 in Chen and Zhao (2017) for the case of V=∇f. When the sectional curvature of the target is nonpositive, we obtain a Liouville theorem for the general V-harmonic maps, as a consequence, any V-harmonic function u, satisfying |u(x)|=o(r(x)), on a complete Riemannian manifold with nonnegative Bakry–Emery–Ricci curvature is a constant. We also give some applications on gradient Ricci solitons and gradient solitons with potential which are solutions to Ricci-harmonic flow.

New Calabi–Bernstein type results in Lorentzian product spaces with density

We investigate the rigidity of complete spacelike hypersurfaces immersed in a weighted Lorentzian product space of the type R1×Pfn, endowed with a weight function f which does not depend on the parameter t∈R and whose Riemannian fiber Pn has nonnegative Bakry–Émery–Ricci tensor. In this direction, supposing that the f-mean curvature is constant and assuming appropriate constraints on the norm of the gradient of f, we prove that such a spacelike hypersurface must be a slice of the ambient space. As application of our investigation, we obtain new Calabi–Bernstein type results concerning entire spacelike graphs constructed over Pn. Our approach is based on an extension for the drift Laplacian of a generalized maximum principle of Akutagawa (1987) and a Liouville-type result due to Pigola et al. (2005).

Generalization of Philippin’s results for the first Robin eigenvalue and estimates for eigenvalues of the bi-drifting Laplacian

In the present paper, we first consider the weighted eigenvalue problem $$\Delta _f u=\lambda _{f}u$$ in M with the Robin boundary condition $$\frac{\partial u}{\partial \nu }+\beta u=0$$ on $$\partial M$$ , where $$(M^n,g,e^{-f})$$ is a compact n-dimensional weighted Riemannian manifold of nonnegative Bakry–Emery Ricci curvature. We derive under some convexity condition of the boundary $$\partial M$$ , an explicit lower bound of the first weighted Robin eigenvalue $$\lambda _{1,f}(\beta )$$ depending only on the geometry of M and the constant $$\beta $$ appearing in the boundary condition. Another new estimate for $$\lambda _{1,f}(\beta )$$ with respect to the first nonzero Neumann eigenvalue $$\mu _{2,f}$$ of the weighted Laplacian $$\Delta _f$$ is also obtained. Furthermore, we provide some lower bounds for the first buckling and clamped plate eigenvalues of the bi-drifting Laplacian on weighted manifolds.

L^p-Analysis of the Hodge\u2013Dirac Operator Associated with Witten Laplacians on Complete Riemannian Manifolds

We prove R-bisectoriality and boundedness of the H^infty -functional calculus in L^p for all 1<p<infty for the Hodge–Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry–Emery Ricci curvature on k-forms.

Rigidity of entire graphs in weighted product spaces with nonnegative Bakry–Émery–Ricci tensor

Abstract We study the rigidity of entire graphs defined over the fiber of a weighted product space whose Bakry–Émery–Ricci tensor is nonnegative. Supposing that the weighted mean curvature is constant and assuming appropriated constraints on the norm of the gradient of the smooth function u which determines such a graph Σ(u), we prove that u is constant. Our proof is based on a formula for the Laplacian of an angle function attached to a hypersurface and on a weakversion of the Omori–Yau generalized maximum principle.

Height estimates and half-space type theorems in weighted product spaces with nonnegative Bakry–Émery–Ricci curvature

We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative Bakry–Emery–Ricci curvature. As applications of our estimates, we obtain half-space type results related to complete noncompact hypersurfaces properly immersed in such an ambient space.

Riemannian metrics on ℝ n and Sobolev-type Inequalities

Poincare-type estimates for a logarithmically concave measure μ on a convex set Ω are obtained. For this purpose, Ω is endowed with a Riemannian metric g in which the Riemannian manifold with measure (Ω, g, μ) has nonnegative Bakry–Emery tensor and, as a corollary, satisfies the Brascamp–Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincare, Hardy, or log-Sobolev type inequalities and other results.

Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space

We prove new sharp Lp, logarithmic, and weak-type inequalities for martingales under the assumption of differential subordination. The Lp estimates are “Feynman–Kac” type versions of Burkholder's celebrated martingale transform inequalities. From the martingale Lp inequalities we obtain that Riesz transforms on manifolds of nonnegative Bakry–Emery Ricci curvature have exactly the same Lp bounds as those known for Riesz transforms in the flat case of Rd. From the martingale logarithmic and weak-type inequalities we obtain similar inequalities for Riesz transforms on compact Lie groups and spheres. Combining the estimates for spheres with Poincaré's limiting argument, we deduce the corresponding results for Riesz transforms associated with the Ornstein–Uhlenbeck semigroup, thus providing some extensions of P.A. Meyer's Lp inequalities.

A Compactness Theorem of the Space of Free Boundary f-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

Let \((M^3,g,e^{-f}d\mu _M)\) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry–Emery Ricci curvature and the boundary \(\partial M\) is strictly f-mean convex. We prove that there exists a properly embedded smooth f-minimal surface \(\Sigma \) in M with free boundary \(\partial \Sigma \) on \(\partial M\). If we further assume that the boundary \(\partial M\) is strictly convex, then we prove that \(M^3\) is diffeomorphic to the 3-ball \(B^3\), and a compactness theorem for the space of properly embedded f-minimal surfaces with free boundary in such \((M^3,g,e^{-f}d\mu _M)\), when the topology of these f-minimal surfaces is fixed.

Heat kernel on smooth metric measure spaces with nonnegative curvature

We derive a local Gaussian upper bound for the $f$-heat kernel on complete smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature, which generalizes the classic Li-Yau estimate. As applications, we obtain a sharp $L_f^1$-Liouville theorem for $f$-subharmonic functions and an $L_f^1$-uniqueness property for nonnegative solutions of the $f$-heat equation, assuming $f$ is of at most quadratic growth. In particular, any $L_f^1$-integrable $f$-subharmonic function on gradient shrinking or steady Ricci solitons must be constant. We also provide explicit $f$-heat kernel for Gaussian solitons.

Foliations of a smooth metric measure space by hypersurfaces with constantf-mean curvature

We study smooth codimension-one foliations F of a smooth metric measure space whose leaves have the same constant f -mean curvature. Firstly, we show that all the leaves of F are f -minimal hypersurfaces when either the smooth metric measure space is compact and has nonnegative Bakry‐ Emery Ricci curvature, or the limit of the ratio of the weighted volume of a geodesic ball B and the weighted area of a geodesic sphere @B vanishes. Secondly, we prove that every leaf of F is strongly f -stable. Lastly, we show that there is no complete proper foliation of the Gaussian space whose leaves have the same constant f -mean curvature. In particular, there are no foliations of R nC1 whose leaves are complete proper self-similar solutions for mean curvature flow.

Monotonicity Formulas for the Bakry–Emery Ricci Curvature

Motivated and inspired by the recent work of Colding (Acta Math 209(2):229–263, 2012) and Colding and Minicozzi (Calc Var Partial Differ Equ 49(3–4):1045–1059, 2014) we derive several families of monotonicity formulas for manifolds with nonnegative Bakry–Emery Ricci curvature, extending the formulas in Colding (2012) and Colding and Minicozzi (2014).

Fundamental Groups of Spaces with Bakry–Emery Ricci Tensor Bounded Below

We first extend Cheeger–Colding’s Almost Splitting Theorem (Ann Math 144:189–237, 1996) 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, 1997), while the second extends a result of Kapovitch and Wilking (Structure of fundamental groups of manifolds with Ricci curvature bounded below, 2011).

Geometry of manifolds with densities

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry–Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry–Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.While some of our results are new even for the gradient Ricci solitons, the novelty here is that only a lower bound of the Bakry–Émery curvature is involved in our analysis.

Harmonic forms on manifolds with non-negative Bakry–Émery–Ricci curvature

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 L 2 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.

Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures

We study the optimal transportation mapping$\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$.Following a classicalapproach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space$M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--Émery tensor provided both $V$ and $W$ are convex.If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space.Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration argumentswe proof some diameter estimates for $M$.

Applications of some elliptic equations in Riemannian manifolds

Let (Mn+1,g) be a compact Riemannian manifold with smooth boundary B and nonnegative Bakry–Emery Ricci curvature. In this paper, we use the solvability of some elliptic equations to prove some estimates of the weighted mean curvature and some related rigidity theorems. As their applications, we obtain some lower bound estimate of the first nonzero eigenvalue of the drifting Laplacian acting on functions on B and some corresponding rigidity theorems.