Let $(M,g)$ be a complete non-compact Riemannian manifold with the $m$-dimensional Bakry-Émery Ricci curvature bounded below by a non-positive constant. In this paper, we give a localized Hamilton-type gradient estimate for the positive smooth bounded solutions to the following nonlinear diffusion equation \[ u_t=Δu-\nablaϕ\cdot\nabla u-au\log u-bu, \] where $ϕ$ is a $C^2$ function, and $a\neq0$ and $b$ are two real constants. This work generalizes the results of Souplet and Zhang (Bull. London Math. Soc., 38 (2006), pp. 1045-1053) and Wu (Preprint, 2008).
We prove a version of differential Harnack inequality for a family of sub-elliptic diffusions on Sasakian manifolds under certain curvature conditions.
We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.
We prove a sharp integral gradient estimate for harmonic functions on noncompact Kähler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds achieving this estimate.
In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature.
In this short note, we study the gradient estimate of positive solutions to Poisson equation and the non-homogeneous heat equation in a compact Riemannian manifold (M^n,g). Our results extend the gradient estimate for positive harmonic functions and positive solutions to heat equations.
In this paper, we prove the local gradient estimate for harmonic functions on complete, noncompact Finsler measure spaces under the condition that the weighted Ricci curvature has a lower bound. As applications, we obtain Liouville type theorem on Finsler manifolds with nonnegative Ricci curvature.
Let $(M, g)$ be an $n$-dimensional complete Riemannian manifold with $Ric(M)\geq-(n-1)Q$, where $Q\geq0$ is a constant. We obtain an interior gradient bound for minimal graphs in $M\times R$ under some technical assumptions. For details, see Theorem 2.
In this paper, we study two kind of L^2 norm preserved non-local heat flows on closed manifolds. We first study the global existence, stability and asymptotic behavior to such non-local heat flows. Next we give the gradient estimates of positive solutions to these heat flows.
We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the Ricci curvature. These estimates imply sharp gradient bounds relating the gradient of an arbitrary solution at given height to that of a symmetric solution on a warped product model space. We also discuss the problem on Finsler manifolds with nonnegative weighted Ricci curvature, and on complete manifolds with bounded geometry, including solutions on manifol…
In this work we derive local gradient and Laplacian estimates of the Aronson-Bénilan and Li-Yau type for positive solutions of porous medium equations posed on Riemannian manifolds with a lower Ricci curvature bound. We also prove similar results for some fast diffusion equations. Inspired by Perelman's work we discover some new entropy formulae for these equations.
We prove a gradient estimate for Donaldson's equation \[ω\wedge(χ+\sqrt{-1}\partial\overline{\partial}φ)^{n-1}=e^F(χ+\sqrt{-1}\partial\overline{\partial}φ)^n\] (and its parabolic analog) on an $n$-dimensional compact Kähler manifold $(M,ω)$ with another Hermitian metric $χ$ directly from the uniform upper bounds for $tr_ωχ_φ$ and Alexandrov-Bakelman-Pucci (ABP) maximum principle.
In this paper, we establish second order estimates for a general class of fully nonlinear equations with linear gradient terms on compact almost Hermitian manifolds. As an application, we first prove the existence of solutions for the Monge-Ampère equation with linear gradient terms for $(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in the almost Hermitian setting. Second, we solve the Monge-Ampère equation and Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a priori estimates for the deformed Hermitian-Yang-Mills equation with supercrit…
Let $(\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup_{x\in M}r^2\left(\oint_{B(x,r)}|Ric^-|^p dV\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac{n}{2}$, when $k(p,1)$ is small enough, certain Li-Yau type gradient bound holds for the positive solutions of the heat equation on geodesic balls $B(O,r)$ in $\M$ with $0<r\leq 1$. Here the assumption that $k(p,1)$ being small allows the situation where the manifolds is collapsing. Recall that in \cite{ZZ}, certain L…
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: $$u_t=ΔF(u),$$ with $F'(u) > 0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME): $$u_t=Δ(u^p),\qquad p>0,$$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of $p$ than that for Aronson-Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for…
We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.
In this note we show how results in \cite{BaudoinBonnefont2016, BaudoinGarofalo2013, CoulhonJiangKoskelaSikora2017} yield the Cheng-Yau estimate on two classes of sub-Riemannian manifolds: Carnot groups and sub-Riemannian manifolds satisfying a generalized curvature-dimension inequality.
Inspired by the study of $V$-static manifold about classification, in this article, we apply the recent results obtained by Freitas and Gomes (Compact gradient Einstein-type manifolds with boundary, 2022) to prove the rigidity results for compact gradient Einstein-type manifolds with nonempty boundary and constant scalar curvature under some suitable pinching conditions. As a special case of gradient Einstein-type manifold, we also give a rigidity result of $(m,ρ)$-quasi-Einstein manifold with boundary.
Studying various functionals and associated gradient ows are known problems in differential geometry. The perpose of this article is to provide a general overview of curvature functionals in Finsler geometry and use their information for introducing different gradient ows on Finsler manifolds.
In this note we present some gradient estimates for the diffusion equation $\partial_t u=Δu-\nabla ϕ\cdot \nabla u $ on Riemannian manifolds, where $ϕ$ is a C^2 function, which generalize estimates of R. Hamilton's and Qi S. Zhang's on the heat equation.
We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized $Γ$-calculus techniques. The advantages and drawbacks of each of these methods are discussed.
In this paper, we will give a horizontal gradient estimate of positive solutions of $Δ_b u = - λu$ on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive pseudo-harmonic functions on Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.
In this paper, motivated by the work of Qi S. Zhang in [28], we derive Li-Yau gradient bounds for positive solutions of the f-heat equation on closed manifolds with Bakry-Emery Ricci curvature bounded below.
The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient $ρ$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient $ρ$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost $η$-Ricci soliton (see Theorem 2).
In this short note, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $$Δu+cu^α=0,$$ where $c, α$ are two real constants and $c\neq 0$.
In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation.
In this note, we show that a nontrivial, compact, degenerate or nondegenerate, gradient Einstein-type manifold of constant scalar curvature is isometric to the standard sphere with a well defined potential function. Moreover, under some geometric assumptions, the noncompact case is also treated. In this case, the main result is that a homogeneous, proper, noncompact, nondegenerate, gradient Einstein-type manifold is an Einstein manifold.
Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.
We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on $(m,0)$ forms, i.e., sections of the canonical bundle of Kähler manifolds, where $m$ is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for the gradient of $(m, 0)$ forms, which involves only the Ricci curvature and the gradient of the scalar curvature.
In this paper we study a large deviation principle of Freidlin-Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.
Using center manifolds and topological degree theory, we construct a new family of complete, $SU(2)$-invariant and steady gradient Ricci solitons on the four-dimensional non-compact cohomogeneity one manifold with group diagram $\mathbb{Z}_4\subset U(1)\subset SU(2)$. We also provide simpler constructions of the existing $U(2)$-invariant steady and complete gradient solitons on the cohomogeneity one manifolds with group diagrams $\mathbb{Z}_n\subset U(1)\subset SU(2)$ for any $n\in \mathbb{N}$, including Appleton's non-collapsed solitons for $n\ge 3$.
In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold $\varrg$, unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen.
We show that the Dirac operator on a spin manifold does not admit $L^2$ eigenspinors provided the metric has a certain asymptotic behaviour and is a warped product near infinity. These conditions on the metric are fulfilled in particular if the manifold is complete and carries a non-complete vector field which outside a compact set is gradient conformal and non-vanishing.
For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to …
We prove that a compact, connected, and oriented 4-dimensional gradient $m$-quasi-Einstein manifold with $m\in [1, \infty]$ which is additionally a spin manifold must satisfy the Hitchin-Thorpe Inequality. We show further that the homeomorphism-type of the universal cover of such a manifold is either $S^4$ or a connected sum of some number of $S^2\times S^2$ when the potential function is nontrivial.
Gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian $Π$-manifolds are studied. It is proved that these objects have constant soliton coefficients. For the soliton under study is shown that the corresponding scalar curvatures of the considered both metrics are equal and constant and its Ricci tensor is a constant multiple of the vertical component. Explicit example of a 3-dimensional para-Sasaki-like Riemannian $Π$-manifold is provided in support of the proved assertions.
We study the gradient flow of the $L^2-$norm of the second fundamental form of smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained for the Willmore flow in Riemannian manifolds, we prove lifespan estimates in terms of the $L^2-$concentration of the second fundamental form of the initial data and we show existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result for the flow and subconvergence to a critical immersion.
Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. It is proved that the Ricci tensor of such a soliton is the vertical component of both B-metrics multiplied by a constant. It is established that gradient almost Ricci-like solitons have constant soliton coefficients. Explicit examples of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures studied are provided.
In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $$ u_t-Δu=au\log u+Vu, \ \ u>0 $$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth function on $M$ with $-ΔV\leq A$ for some positive constant $A$. This heat equation is a basic evolution equation and it can be considered as the negative gradient heat flow to $W$-functional (introduced by G.Perelman), which is the Log-Sobolev inequalities on the Riemannian manifold and $V$ correspond…
In this paper, by employ the cutoff function and the maximum principle, some Hamilton-Souplet-Zhang type gradient estimates for porous medium type equation are deduced. As a special case, an Hamilton-Souplet-Zhang type gradient estimates of the heat equation is derived which is different from the result of Souplet-Zhang. Furthermore, our results generalize those of Zhu. As application, some Livillous theorems for ancient solution are derived.
In this paper, we classify the compact locally homogeneous non-gradient $m$-quasi Einstein 3-manifolds. Along the way, we prove that given a compact quotient of a Lie group of any dimension that is $m$-quasi Einstein, the potential vector field $X$ must be left invariant and Killing. We also classify the nontrivial $m$-quasi Einstein metrics that are a compact quotient of be the product of two Einstein metrics. We also show that $S^1$ is the only compact manifold of any dimension which admits a metric which is nontrivially $m$-quasi Einstein and Einstein.
Inspired by Wilkin's work [23, 24] on Morse theory for the moduli space of Higgs bundles, we study the moduli space of gauged holomorphic maps by a heat flow approach in the spirit of Atiyah and Bott in a series of papers. In this paper, applying the method of Hong [9], we establish the global existence of smooth solutions of the gradient ow equations of the vortex functional over a compact Kahler manifold.
For positive $p$-harmonic functions on Riemannian manifolds, we derive a gradient estimate and Harnack inequality with constants depending only on the lower bound of the Ricci curvature, the dimension $n$, $p$ and the radius of the ball on which the function is defined. Our approach is based on a careful application of the Moser iteration technique and is different from Cheng-Yau's method employed by Kostchwar and Ni, in which a gradient estimate for positive $p$-harmonic functions is derived under the assumption that the sectional curvature is bounded from below.
The purpose of this article is to study the geometry of gradient almost Yamabe solitons immersed into warped product manifolds $I\times_{f}M^{n}$ whose potential is given by the height function from the immersion. First, we present some geometric rigidity on compact solitons due to a curvature condition on the warped product manifold. In the sequel, we investigate conditions for the existence of totally geodesic, totally umbilical and minimal solitons. Furthermore, in the scope of constant angle immersions, a classification of rotational gradient almost Yamabe soliton immersed into $\mathbb{R}…
On an $n$-dimensional complete manifold $M$, consider an $h$-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and $dh/du>0$, then the manifold $M$ is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. Moreover, if the dimension of $M$ is four, the metric $g$ is conformally flat.
In this paper, we prove sharp gradient estimates for positive solutions to the weighted heat equation on smooth metric measure spaces with compact boundary. As an application, we prove Liouville theorems for ancient solutions satisfying the Dirichlet boundary condition and some sharp growth restriction near infinity. Our results can be regarded as a refinement of recent results due to Kunikawa and Sakurai.
