Laplacian Comparison Theorem Research Articles

A new Laplacian comparison theorem and its applications on Finsler manifold

A new Laplacian comparison theorem and its applications on Finsler manifold

New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds

Let L = Δ − ⟨ ∇ ϕ , ∇ · ⟩ $L=\Delta -\langle \nabla \phi , \nabla \cdot \rangle$ be a symmetric diffusion operator with an invariant measure μ ( d x ) = e − ϕ ( x ) m ( d x ) $\mu (\mathrm{d}x)=e^{-\phi (x)}\mathfrak {m} (\mathrm{d}x)$ on a complete non-compact smooth Riemannian manifold ( M , g ) $(M,g)$ with its volume element m = vol g $\mathfrak {m} =\text{\rm vol}_g$ , and ϕ ∈ C 2 ( M ) $\phi \in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with CD ( K , m ) ${\rm CD}(K, m)$ -condition for m ⩽ 1 $m\leqslant 1$ and a continuous function K $K$ . As consequences, we give the optimal conditions on m $m$ -Bakry–Émery Ricci tensor for m ⩽ 1 $m\leqslant 1$ such that the (weighted) Myers' theorem, Bishop–Gromov volume comparison theorem, stochastic completeness and Feller property of L $L$ -diffusion processes hold on weighted complete Riemannian manifolds. Some of these results were well studied for m $m$ -Bakry–Émery Ricci curvature for m ⩾ n $m\geqslant n$ (Li, J. Math. Pures Appl. (9) 84 (2005), 1295–1361; Lott, Comment. Math. Helv. 78 (2003), 865–883; Qian, Q. J. Math. 48 (1987), 235–242; Wei and Wylie, J. Differential Geom. 83 (2009), 377–405) or m = 1 $m=1$ (Wylie, Trans. Amer. Math. Soc. 369 (2017), 6661–6681; Wylie and D. Yeroshkin, Preprint). When m < 1 $m<1$ , our results are new in the literature.

Laplacian comparison theorem on Riemannian manifolds with modified $m$-Bakry-Emery Ricci lower bounds for $m\leq1$

In this paper, we prove a Laplacian comparison theorem for non-symmetric diffusion operator on complete smooth $n$-dimensional Riemannian manifold having a lower bound of modified $m$-Bakry-Émery Ricci tensor under $m\leq 1$ in terms of vector fields. As consequences, we give the optimal conditions for modified $m$-Bakry-Émery Ricci tensor under $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, Cheng's maximal diameter theorem, and the Cheeger-Gromoll type splitting theorem hold. Some of these results were well-studied for $m$-Bakry-Émery Ricci curvature under $m\geq n$ ([19, 21, 27, 33]) or $m=1$ ([34, 35]) if the vector field is a gradient type. When $m<1$, our results are new in the literature.

Comparison Theorems on Weighted Finsler Manifolds and Spacetimes with ϵ-Range

Abstract We establish the Bonnet–Myers theorem, Laplacian comparison theorem, and Bishop–Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with ϵ-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie–Yeroshkin and Kuwae–Li.

Rigidity phenomena on lower N-weighted Ricci curvature bounds with ε-range for nonsymmetric Laplacian

Lu, Minguzzi, and Ohta introduced the notion of a lower N-weighted Ricci curvature bound with ε-range and derived several comparison geometric estimates from a Laplacian comparison theorem for a weighted Laplacian. The aim of this paper is to investigate various rigidity phenomena for the equality case of their comparison geometric results. We will obtain rigidity results concerning the Laplacian comparison theorem, diameter comparisons, and volume comparisons. We also generalize their works for a nonsymmetric Laplacian induced from vector field potential.

Dualities in comparison theorems and bundle-valued generalized harmonic forms on noncompact manifolds

We observe, utilize dualities in differential equations and differential inequalities (see Theorem 2.1), dualities between comparison theorems in differential equations (see Theorems E and 2.2), and obtain dualities in ‘swapping’ comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II (see Theorems 2.3 and 2.4) and lead to comparison theorems in Riemannian geometry (see Theorems 2.5 and 2.8) with analytic, geometric, PDE’s and physical applications. In particular, we prove Hessian comparison theorems (see Theorems 3.1–3.5) and Laplacian comparison theorems (see Theorems 2.6, 2.7 and 3.1–3.5) under varied radial Ricci curvature, radial curvature, Ricci curvature and sectional curvature assumptions, generalizing and extending the work of Han-Li-Ren-Wei (2014) and Wei (2016). We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship (see Theorem 5.4). These provide tools in extending the notion, integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms, introducing Condition W for bundle-valued differential forms, and proving the duality theorem and the unity theorem, generalizing the work of Andreotti and Vesentini (1965) and Wei (2020). We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature, generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds, the embedding theorem for weighted Sobolev spaces of functions on manifolds, geometric differential-integral inequalities, generalized sharp Hardy type inequalities on Riemannian manifolds, monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles, such as F-Yang-Mills fields (when F is the identity map, they are Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds, Liouville type theorems for F - harmonic maps (when $$F\left(t \right) = {1 \over p}{\left({2t} \right)^{{p \over 2}}},p > 1$$ , they become p-harmonic maps or harmonic maps if p = 2), and Dirichlet problems on starlike domains for vector bundle valued differential 1-forms and {tF}-harmonic maps (see Theorems 4.1, 7.3–7.7, 8.1, 9.1–9.3, 10.1, 11.2, 12.1 and 12.2), generalizing the work of Caffarelli et al. (1984) and Costa (2008), in which M = ℝn and its radial curvature K(r) = 0, the work of Wei and Li (2009), Chen et al. (2011, 2014), Dong and Wei (2011), Wei (2020) and Karcher and Wood (1984), etc. The boundary value problem for bundle-valued differential 1-forms is in contrast to the Dirichlet problem for p-harmonic maps to which the solution is due to Hamilton (1975) for the case p = 2 and RiemN ⩽ 0, and Wei (1998) for 1 < p < ∞.

Brownian Motions and Heat Kernel Lower Bounds on Kähler and Quaternion Kähler Manifolds

Abstract We study the radial parts of the Brownian motions on Kähler and quaternion Kähler manifolds. Thanks to sharp Laplacian comparison theorems, we deduce as a consequence a sharp Cheeger–Yau-type lower bound for the heat kernels of such manifolds and also sharp Cheng’s type estimates for the Dirichlet eigenvalues of metric balls.

Comparison Theorems in Finsler Geometry with Integral Weighted Ricci Curvature Bounds and Their Applications

We establish the Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under integral weighted Ricci curvature bounds. As their applications, we obtain some results on integral weighted Ricci curvature and topology for Finsler manifolds. We also obtain some results on the upper bound of first eigenvalue for Finsler manifolds under integral weighted Ricci curvature bounds.

An Extension Theorem for Weighted Ricci Curvature on Finsler Manifolds

Let (M,F) be a forward complete and connected Finsler manifold of dimensional n ≥2 . In this study, we extend Wan’s extension theorem in Riemannian manifolds to Finsler manifolds by using the weighted Ricci curvature RicN bounded below. The proof of theorem is obtained by the Laplacian comparison theorem on Finsler manifolds and the excess function.

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical Laplacian comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical Laplacian comparison theorems for the Riemannian distances approximations. As a corollary we prove that, under suitable curvature conditions, sub-Riemannian Sasakian spaces are actually limits of Riemannian spaces satisfying a uniform measure contraction property.

Pseudo-harmonic Maps from Complete Noncompact Pseudo-Hermitian Manifolds to Regular Balls

In this paper, we give an estimate of sub-Laplacian of Riemannian distance functions in pseudo-Hermitian geometry which plays a similar role as Laplacian comparison theorem in Riemannian geometry, and deduce a prior horizontal gradient estimate of pseudo-harmonic maps from pseudo-Hermitian manifolds to regular balls of Riemannian manifolds. As an application, Liouville theorem is established under the conditions of nonnegative pseudo-Hermitian Ricci curvature and vanishing pseudo-Hermitian torsion. Moreover, we obtain the existence of pseudo-harmonic maps from complete noncompact pseudo-Hermitian manifolds to regular balls of Riemannian manifolds.

Comparison theorems on Finsler manifolds with weighted Ricci curvature bounded below

We obtain the Laplacian comparison theorem and the Bishop-Gromov comparison theorem on a Finsler manifold with the weighted Ricci curvature Ric∞ bounded below. As applications, we prove that if the weighted Ricci curvature Ric∞ is bounded below by a positive number, then the manifold must have finite fundamental group, and must be compact if the distortion is also bounded. Moreover, we give the Calabi-Yau linear volume growth theorem on a Finsler manifold with nonnegative weighted Ricci curvature.

Gradient Estimates for the Positive Solutions of $$\mathfrak {L}u=0$$ L u = 0 on Self-Shrinkers

In this paper, we investigate the positive solutions of $$\mathfrak {L}u=0$$ on a self-shrinker. First, we prove a global gradient estimate for the positive solutions, and obtain a strong Liouville theorem. Then by the generalized Laplacian comparison theorem for the distance function on a self-shrinker, we derive a local gradient estimate for the positive solutions. At last, we collect some applications of the local gradient estimate for the positive solutions on self-shrinkers.

Bishop and Laplacian comparison theorems on Sasakian manifolds

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for a natural sub-Riemannian structure defined on Sasakian manifolds. This generalizes the earlier work for the three dimensional case.

Comparison Theorems and Their Applications on Kähler Finsler Manifolds

We obtain complex Hessian and Laplacian comparison theorems on Kahler Finsler manifolds with holomorphic bisectional curvature bounded from below. As applications, we prove Bishop type volume comparison theorem and Myers type theorem for Kahler Finsler manifolds. Moreover, we also obtain some comparison theorems for the first eigenvalue of Finsler–Laplacian and give a sharp upper estimate of the bottom of the spectrum for Kahler Finsler manifolds whose holomorphic bisectional curvature is bounded from below by $$-\,1$$.

SUB-RIEMANNIAN RICCI CURVATURES AND UNIVERSAL DIAMETER BOUNDS FOR 3-SASAKIAN MANIFOLDS

For a fat sub-Riemannian structure, we introduce three canonical Ricci curvatures in the sense of Agrachev–Zelenko–Li. Under appropriate bounds we prove comparison theorems for conjugate lengths, Bonnet–Myers type results and Laplacian comparison theorems for the intrinsic sub-Laplacian. As an application, we consider the sub-Riemannian structure of 3-Sasakian manifolds, for which we provide explicit curvature formulas. We prove that any complete 3-Sasakian structure of dimension$4d+3$, with$d&gt;1$, has sub-Riemannian diameter bounded by$\unicode[STIX]{x1D70B}$. When$d=1$, a similar statement holds under additional Ricci bounds. These results are sharp for the natural sub-Riemannian structure on$\mathbb{S}^{4d+3}$of the quaternionic Hopf fibrations:$$\begin{eqnarray}\mathbb{S}^{3}{\hookrightarrow}\mathbb{S}^{4d+3}\rightarrow \mathbb{HP}^{d},\end{eqnarray}$$whose exact sub-Riemannian diameter is$\unicode[STIX]{x1D70B}$, for all$d\geqslant 1$.

Comparison theorems on smooth metric measure spaces with boundary

Abstract In this paper we study smooth metric measure spaces with boundary via the Bakry–Émery curvature and the weighted mean curvature of the boundary. We establish the weighted Laplacian comparison theorems and the upper bound estimates of the distance from any point of the manifold to its boundary. As applications, we derive lower bound estimates for the first Dirichlet eigenvalue.

Volumes and limits of manifolds with Ricci curvature and mean curvature bounds

We consider smooth Riemannian manifolds with nonnegative Ricci curvature and smooth boundary. First we prove a global Laplacian comparison theorem in the barrier sense for the distance to the boundary. We apply this theorem to obtain volume estimates of the manifold and of regions of the manifold near the boundary depending upon an upper bound on the area and on the inward pointing mean curvature of the boundary. We prove that families of oriented manifolds with uniform bounds of this type are compact with respect to the Sormani–Wenger Intrinsic Flat (SWIF) distance.

Comparison theorems for manifolds with mean convex boundary

Let Mn be an n-dimensional Riemannian manifold with boundary ∂M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k ∈ ℝ, we give a sharp estimate of the upper bound of ρ(x) = d (x, ∂M), in terms of the mean curvature bound of the boundary. When ∂M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kähler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kähler manifold and also estimate the first eigenvalue of the real Laplacian.

A generalized Omori–Yau maximum principle in Finsler geometry

We obtain some new Laplacian comparison theorems on Finsler manifolds with the (weighted) Ricci curvature bounded from above (resp. below) by a smooth function. Then, under some new sufficient conditions, we generalize the Omori–Yau type maximum principle in the Finsler setting.