Laplacian Comparison Theorem Research Articles

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.

Some comparison theorems and their applications in Finsler geometry

Abstract By using arbitrary volume forms, we establish Laplacian comparison theorems for Finsler manifolds under certain curvature conditions. As applications, some volume comparison theorems and Mckean type eigenvalue estimates of Finsler manifolds are obtained. Moreover, we also generalize Calabi-Yau’s linear volume growth theorem, and Milnor’s results on curvature and the fundamental group to the Finsler setting. MSC: 53C60, 53B40.

On the Curvature and Heat Flow on Hamiltonian Systems

Abstract We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation.We prove appropriate generalizations of the Bochner-Weitzenböck formula and Laplacian comparison theorem, and study the heat flow.

Bishop and Laplacian Comparison Theorems on Three-Dimensional Contact Sub-Riemannian Manifolds with Symmetry

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three-dimensional contact sub-Riemannian manifolds with symmetry.

Some generalized comparison results in Finsler geometry and their applications

In this paper, we generalize the Hessian comparison theorems and Laplacian comparison theorems described in [16, 18], then give some applications under various curvature conditions.

A probabilistic approach to the maximal diameter theorem

AbstractUnder a positive curvature and a finite dimension in terms of the Bakry‐Émery tensor on a Riemannian manifold, the Bonnet‐Myers type diameter bound and the rigidity theorem are extended. The corresponding second order generator need not be symmetrizable. The proof is based on the Laplacian comparison theorem and stochastic analysis of radial processes.

Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori–Yau maximum principle for certain elliptic operators.

Spectrum of the Laplacian on quaternionic Kähler manifolds

Spectrum of the Laplacian on quaternionic Kähler manifolds

Comparison theorems in Finsler geometry and their applications

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under various curvature conditions. As applications, we derive McKean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize to Finsler manifolds a result on fundamental groups due to Milnor.

AdS Geometry, Projective Embedded Coordinates and Associated Isometry Groups

This work is intended to investigate the geometry of anti-de Sitter spacetime (AdS), from the point of view of the Laplacian Comparison Theorem (LCT), and to give another description of the hyperbolical embedding standard formalism of the de Sitter and anti-de Sitter spacetimes in a pseudoeuclidean spacetime. After Witten proved that general relativity is a renormalizable quantum system in (1+2) dimensions, it is possible to point out few interesting motivations to investigate AdS spacetime. A lot of attempts were made to generalize the gauge theory of gravity in (1+2) dimensions to higher ones. The first one was to enlarge the Poincaré group of symmetries, supposing an AdS group symmetry, which contains the Poincaré group. Also, the AdS/CFT correspondence asserts that a maximal supersymmetric Yang–Mills theory in four-dimensional Minkowski spacetime is equivalent to a type IIB closed superstring theory. The 10-dimensional arena for the type IIB superstring theory is described by the product manifold S 5× AdS, an impressive consequence that motivates the investigations about the AdS spacetime in this paper, together with the de Sitter spacetime. Classical results in this mathematical formulation are reviewed in a more general setting together with the isometry group associated to the de Sitter spacetime. It is known that, out of the Friedmann models that describe our universe, the Minkowski, de Sitter, and anti-de Sitter spacetimes are the unique maximally isotropic ones, so they admit a maximal number of conservation laws and also a maximal number of Killing vectors. In this paper it is shown how to reproduce some geometrical properties of AdS, from the LCT in AdS, choosing suitable functions that satisfy basic properties of Riemannian geometry. We also introduce and discuss the well-known embedding of a four-sphere and a four-hyperboloid in a five-dimensional pseudoeuclidean spacetime, reviewing the usual formalism of spherical embedding and the way how it can retrieve the Robertson–Walker metric. With the choice of the de Sitter metric static frame, we write the so-called reduced model in suitable coordinates. We assume the existence of projective coordinates, since de Sitter spacetime is orientable. From these coordinates, obtained when stereographic projection of the de Sitter four-hemisphere is done, we consider the Beltrami geodesic representation, which gives a more general formulation of the seminal full model described by Schr“odinger, concerning the” geometry and the topology of de Sitter spacetime. Our formalism retrieves the classical one if we consider the metric terms over the de Sitter splitting on Minkowski spacetime. From the covariant derivatives we find the acceleration of moving particles, Killing vectors and the isometry group generators associated to de the Sitter spacetime.

Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds

Let L = Δ − ∇ ϕ ⋅ ∇ be a symmetric diffusion operator with an invariant measure μ ( d x ) = e − ϕ ( x ) d x on a complete non-compact Riemannian manifold M. We give the optimal conditions on “the m-dimensional Ricci curvature associated with L” so that various Liouville theorems hold for L-harmonic functions, and that the heat semigroup P t = e t L has the C 0 -diffusion property and is unique in L 1 ( M , μ ) . As applications, we give the optimal conditions for the uniqueness of the positive L-invariant measure and the L 1 -uniqueness of the intrinsic Schrödinger operators on complete non-compact Riemannian manifolds. We also give a criterion for the finiteness of the total mass of the L-invariant measure and establish the Calabi–Yau volume growth theorem for the L-invariant measure on complete Riemannian manifolds on which “the m-dimensional Ricci curvature associated with L” is non-negative. This leads us to prove that if M is a complete Riemannian manifold with a finite L-invariant measure for which the associated m-dimensional Ricci curvature is non-negative, then M is compact. Moreover, we obtain an upper bound diameter estimate of such Riemannian manifolds by using the dimension of L, the total μ-volume of M and the upper bound of the μ-volume of geodesic balls of a fixed radius. Finally, using the variational formulae in Riemannian geometry, we give a new proof of the Bakry–Qian generalized Laplacian comparison theorem.

Laplacian comparison and sub-mean-value theorem for multiplier Hermitian manifolds

In this note, we study the Laplacian comparison theorem and the sub-mean-value theorem for a special type of Hermitian manifolds called multiplier Hermitian manifolds. By conformal change of the metrics, this covers much wider objects than in the case of ordinary Kahler manifolds.

A Laplacian comparison theorem and its applications

A Laplacian comparison theorem is given. As applications we show a volume comparison theorem and a criterion for the hyperbolicity of Riemannian manifolds.

A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold

A Laplacian comparison theorem and function theoretic properties of a complete Riemannian manifold

Ricci curvature and convergence of Lipschitz functions

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and distributional Laplacian comparison theorem on limit spaces of Riemannian manifolds.

A general Schwarz lemma for almost-Hermitian manifolds

We prove a version of Yau's Schwarz Lemma for general almost-complex manifolds equipped with Hermitian metrics. This requires an extension to this setting of the Laplacian comparison theorem. As an application we show that the product of two almost-complex manifolds does not admit any complete Hermitian metric with bisectional curvature bounded between two negative constants that satisfies some additional mild assumptions.