Neckpinch — Harnack inequalities

Pseudolocality of G₂-Laplacian Flow

Wed, 28 Oct 2026 00:00:00 GMT

Weiyong He, Longhuan Yan — In this paper, we give a new proof of Perelman's pseudolocality theorem for Ricci flow, which does not rely on Perelman's Harnack-type inequality. The key step is an estimate relating initial local μ-entropy to pointed Nash entropy at later spacetime points. The argument also works for other reasonable geometric flows, like the G₂-Laplacian flow, at a scale depending on its deviation from Ricci flow and its scalar curvature lower bound. As a consequence, we can also prove the quantitative rigidity results concerning pointed Nash entropy due to Hein-Naber and Bamler for such geometric flows. In the appendix, we also show that the other results in Bamler's work (arXiv:2008.07093), when suitably modified, remain valid for Laplacian flow with bounded torsion. These include log-Sobolev/Poincaré inequality for conjugate heat kernel measure, heat kernel estimates, and dependence of the pointed Nash entropy on its basepoint.

Type-II Singularities of Immersed Locally Convex Curves Evolving under Generalized Anisotropic Gage Area-Preserving Flows with Slow Curvature Speeds

Mon, 26 Oct 2026 00:00:00 GMT

Ze-Yu Ye, Xiao-Liu Wang — In this paper, we study the singularity behavior of immersed locally convex closed plane curves evolving under generalized anisotropic Gage area-preserving flows with slow curvature speeds of the form κ_σ^α, where 0<α≤ 1. Here κ_σ denotes the anisotropic curvature associated with a prescribed anisotropy function σ. We prove that every finite-time blow-up is of Type-II provided that the Euclidean length of the evolving curve remains uniformly bounded away from zero. This result holds without any symmetry assumptions on either the initial curve or the anisotropy. When the initial signed area is nonzero, this length lower bound is automatically satisfied, and thus any finite-time singularity is necessarily of Type-II. By means of Hamilton's rescaling method combined with a differential Harnack estimate, we show that along a suitable blow-up subsequence, the rescaled curves at the selected times converge locally to the profile of a translating solution of the corresponding anisotropic curvature flow V=κ_σ^α. We also establish sufficient conditions for finite-time singularity formation in terms of the initial signed area and anisotropic length. Moreover, we construct locally convex initial curves arbitrarily close to the Wulff curve for which a finite-time Type-II singularity occurs.

Higher regularity of the inverse anisotropic mean curvature flow

Thu, 02 Jul 2026 00:00:00 GMT

Chaoqun Gao, Yong Wei, Rong Zhou — We prove an anisotropic analogue of the higher regularity theorem of Huisken and Ilmanen for inverse mean curvature flow. For an arbitrary smooth Minkowski norm, we first prove a Huisken–Ilmanen type Harnack estimate for smooth closed strictly star-shaped solutions. We then construct global smooth solutions starting from C¹ strictly star-shaped hypersurfaces with bounded nonnegative weak anisotropic mean curvature. Combining this construction with the asymptotic theory for weak inverse anisotropic mean curvature flow, we show that weak solutions starting from bounded smooth initial sets become smooth outside a compact set.

Distance between minimal surfaces and flows

Mon, 11 May 2026 00:00:00 GMT

Tobias Holck Colding, William P. Minicozzi — We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.

Lectures on curve shortening flow

Thu, 02 Apr 2026 00:00:00 GMT

Robert Haslhofer — The curve shortening flow is a geometric heat equation for curves and provides an accessible setting to illustrate many important concepts from nonlinear partial differential equations, including maximum principle estimates, monotonicity formulas, Harnack inequalities and blowup analysis. All these techniques will be combined to give an exposition of Huisken's proof of Grayson's beautiful theorem that the curve shortening flow shrinks any closed embedded curve in the plane to a round point.

PIC1 pinched manifolds are flat or compact

Mon, 23 Mar 2026 00:00:00 GMT

Alix Deruelle, Man-Chun Lee, Felix Schulze, Miles Simon, Peter M. Topping — Hamilton's pinching conjecture, that three-dimensional complete non-compact manifolds with pinched Ricci curvature are flat, has recently been resolved using Ricci flow. In this paper we prove a direct analogue of that result in all dimensions. In order to do so we develop a lifting technique that allows us to handle manifolds that are collapsed at infinity. This new method also gives an alternative way of handling collapsed manifolds in the known three-dimensional case. As part of this approach, we prove a Ricci flow curvature estimate of a type that would normally be derived from the Harnack inequality, but without requiring the strong curvature positivity hypothesis demanded by Harnack. We give an improved gap theorem as a further application.

The Harnack inequality without convexity for curve shortening flow

Tue, 20 Jan 2026 00:00:00 GMT

Arjun Sobnack, Peter M. Topping — In 1995, Hamilton introduced a Harnack inequality for convex solutions of the mean curvature flow. In this paper we prove an alternative Harnack inequality for curve shortening flow, i.e. one-dimensional mean curvature flow, that does not require any assumption of convexity. For an initial proper curve in the plane whose ends are radial lines but which is otherwise arbitrarily wild, we use the Harnack inequality to give an explicit time by which the curve shortening flow evolution must become graphical. This gives a new instance of delayed parabolic regularity. The Harnack inequality also gives estimates describing how a polar graphical flow with radial ends settles down to an expanding solution. Finally, we relate our Harnack inequality to Hamilton's by identifying a pointwise curvature estimate implied by both Harnack inequalities in the special case of convex flows.

A Dynamical Approach to the Berezin-Li-Yau Inequality

Mon, 01 Dec 2025 00:00:00 GMT

Anton Alexa — We develop a dynamical method for proving the sharp Berezin-Li-Yau inequality. The approach is based on the volume-preserving mean curvature flow and a new monotonicity principle for the Riesz mean R_Λ(Ωₜ). For convex domains we show that R_Λ is monotone non-decreasing along the flow. The key input is a geometric correlation inequality between the boundary spectral density Q_Λ and the mean curvature H, established in all dimensions: in d=2 via a near-disk Fourier analysis, and in d≥ 3 via the boundary Weyl expansion together with a local spectral rigidity argument near the ball, with a first-zero exclusion principle closing the global step. Since the flow converges smoothly to the ball, the monotonicity implies the sharp Berezin-Li-Yau bound for every smooth convex domain. As an application, we obtain a sharp dynamical Cesàro-Pólya inequality for eigenvalue averages.

Harnack Inequality for f-Mean Curvature Flow

Fri, 28 Nov 2025 00:00:00 GMT

Xiang-Dong Li, Qi Yan — In this paper, we prove a Li-Yau-Hamilton type Harnack estimate for the f-mean curvature flow in Euclidean space, which can be viewed as a gradient flow of the weighed area functional with the measure density function e^(−f).

Monotonicity of Perelman 𝓦-Entropy of Mean Curvature Flow

Fri, 28 Nov 2025 00:00:00 GMT

Xiang-Dong Li, Qi Yan — In this paper, we study Perelman' s 𝓦 entropy for mean curvature flow in ℝⁿ⁺¹. Analogously to Perelman's 𝓦-entropy defined for Ricci flow, K. Ecker in defined a functional 𝓦 for the mean curvature flow in ℝⁿ⁺¹ and the region it encloses, and made the conjecture that this functional is monotonically increasing in time. We modify K. Ecker's definition and, using Hamilton's Harnack inequality for mean curvature flow, prove that our redefined 𝓦-entropy is monotonically decreasing in time. Additionally, we provide a rigidity theorem for this 𝓦-entropy.

Mean curvature flow into an ambient Riemannian manifold evolving by Ricci flow coupled with harmonic map heat flow

Mon, 27 Oct 2025 00:00:00 GMT

José N. V. Gomes, Matheus Hudson, Carlos M. de Sousa — The main objective of this article is to study the mean curvature flow into an ambient compact smooth manifold M with boundary and with a Riemannian metric that evolves by a self-similar solution of the Ricci flow coupled with the harmonic map heat flow of a map from M to a Riemannian manifold N. In this context, we address a functional associated with this flow and calculate its variation along parameters that preserve the weighted volume measure. An extension of Hamilton's differential Harnack expression appears by considering the boundary of M evolving by mean curvature flow, which must vanish on the gradient steady soliton case. Next, we obtain a Huisken monotonicity-type formula for the mean curvature flow in the proposed background. We also show how to construct a family of mean curvature solitons and establish a characterization of such a family.

On a parabolic curvature lower bound generalizing Ricci flows

Mon, 07 Jul 2025 00:00:00 GMT

Marco Flaim, Erik Hupp — Optimal transport plays a major role in the study of manifolds with Ricci curvature bounded below. Some results in this setting have been extended to super Ricci flows, revealing a unified approach to analysis on Ricci nonnegative manifolds and Ricci flows. However we observe that the monotonicity of Perelman's functionals (𝓕, 𝓦, reduced volume), which hold true for Ricci flows and Ricci nonnegative manifolds, cannot be strictly generalized to super Ricci flows. In 2010 Buzano introduced a condition which still generalizes Ricci flows and Ricci nonnegative manifolds, and on which Perelman's monotonicities do hold. We provide characterizations of this condition using optimal transport and understand it heuristically as Ricci nonnegativity of the space-time. This interpretation is consistent with its equivalence to Ricci nonnegativity on Perelman's infinite dimensional manifold. More precisely, we prove that for smooth evolutions of Riemannian manifolds, this condition is equivalent to a Bochner inequality (resembling Perelman's Harnack inequality but for the forward heat flow), a gradient estimate for the heat flow, a Wasserstein contraction along the adjoint heat flow, the convexity of a modified entropy along Wasserstein geodesics, and an Evolutionary Variational Inequality (EVI). The optimal transport statements use Perelman's L distance as cost, as first studied on Ricci flows by Topping and by Lott. We also consider dimensionally improved and weighted versions of these conditions. The dimensional Bochner inequality and all gradient estimates for the forward heat equation, along with the EVIs, appear to be new even for general Ricci flows, and are related to the Hamiltonian perspective on the L distance. Most of our proofs do not use tensor calculus or Jacobi fields, suggesting the possibility of future extensions to more singular settings.

Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow

Thu, 05 Jun 2025 00:00:00 GMT

Juanling Lu, Yu Zheng — In this paper, we establish Li-Yau-type and Hamilton-type estimates for positive solutions to the heat equation associated with the generalized Ricci flow, under a less stringent curvature condition. Compared with [25] and [35], these estimates generalize the results in Ricci flow to this new flow under the weaker Ricci curvature bounded assumption. As an application, we derive the Harnack-type inequalities in spacetime and find the monotonicity of one parabolic frequency for positive solutions of the heat equation under bounded Ricci curvature.

On Perelman's W-entropy and Shannon entropy power for super Ricci flows on metric measure spaces

Tue, 06 May 2025 00:00:00 GMT

Xiang-Dong Li — In this paper, we extend Perelman's W-entropy formula and the concavity of the Shannon entropy power from smooth Ricci flow to super Ricci flows on metric measure spaces. Moreover, we prove the Li-Yau-Hamilton-Perelman Harnack inequality on super Ricci flows. As a significant application, we prove the equivalence between the volume non-local collapsing property and the lower boundedness of the W-entropy on RCD(0, N) spaces. Finally, we use the W-entropy to study the logarithmic Sobolev inequality with optimal constant on super Ricci flows on metric measure spaces.

Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

Tue, 11 Feb 2025 00:00:00 GMT

Junyoung Park — The goal of this paper is to relax convexity assumption on some classical results in mean curvature flow. In the first half of the paper, we prove a generalized version of Hamilton's differential Harnack inequality which holds for mean convex solutions to mean curvature flow with a lower bound on (λ₁)/H where λ₁ is the smallest principal curvature. Then, we use classical maximum principle to provide several characterizations of family of shrinking spheres for closed, mean convex ancient solution to mean curvature flow with a lower bound on (λ₁ + .. + λₖ)/H for some 1 ≤ k ≤ d-1, where λ₁ ≤ λ₂ ≤ .. ≤ λ_d are the principal curvatures.

Geometric flows and space-periodic solitons on the light-cone

Sun, 09 Feb 2025 00:00:00 GMT

Yun Yang — This paper investigates curve flows on the light-cone in the 3-dimensional Minkowski space. We derive the Harnack inequality for the heat flow and present a detailed classification of space-periodic solitons for a third-order curvature flow. The nontrivial periodic solutions to this flow are expressed in terms of the Jacobi elliptic sine function. Additionally, the closed soliton solutions form a family of transcendental curves, denoted by x_(p,q), which are characterized by a rotation index p and close after q periods of their curvature functions. The ratio p/q satisfies p/q ∈ (√(2/3), 1), where p and q are relatively prime positive integers. Guided by the classification process, we obtain the analytic solutions to a second-order nonlinear ordinary differential equation.

Log gradient estimates for heat type equations on manifolds

Tue, 30 Jul 2024 00:00:00 GMT

Qi S. Zhang — In this short survey paper, we first recall the log gradient estimates for the heat equation on manifolds by Li-Yau, R. Hamilton and later by Perelman in conjunction with the Ricci flow. Then we will discuss some of their applications and extensions focusing on sharp constants and improved curvature conditions.

Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers

Tue, 02 Jul 2024 00:00:00 GMT

Jacob Bernstein — We show the (normalized) Li-Yau conformal volume of a self-shrinker of mean curvature flow in Euclidean space bounds its Colding-Minicozzi entropy from below. This bound is independent of codimension and sharp on planes. As an application we verify a conjecture of Colding-Minicozzi about the entropy of closed self-shrinkers of arbitrary codimension for self-shrinkers that are topologically two-dimensional real projective planes. As part of the proof we introduce two auxiliary functionals which we call stable conformal volume and virtual entropy which should be of independent interest.

Metric Measure Spaces and Synthetic Ricci Bounds – Fundamental Concepts and Recent Developments

Wed, 24 Apr 2024 00:00:00 GMT

Karl-Theodor Sturm — Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this survey, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the curvaturedimension condition formulated in functional-analytic terms by Bakry-Èmery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions: i) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for singular spaces, ii) second order calculus, upper Ricci bounds, and transformation formulas, iii) distribution-valued Ricci bounds which e.g. allow singular effects of non-convex boundaries to be taken into account.

Gradient estimation of a generalized non-linear heat type equation along Super-Perelman Ricci flow on weighted Riemannian manifolds

Wed, 24 Apr 2024 00:00:00 GMT

Yanlin Li, Abimbola Abolarinwa, Suraj Ghosh, Shyamal Kumar Hui — In this article we derive gradient estimation for positive solution of the equation \beginequation* (\partial_t-Δ_f)u = A(u)p(x,t) + B(u)q(x,t) + \mathcalG(u) \endequation* on a weighted Riemannian manifold evolving along the (k,m) super Perelman-Ricci flow \beginequation* \frac{\partial g}{\partial t}(x,t)+2Ric_f^m(g)(x,t)\ge -2kg(x,t). \endequation* As an application of gradient estimation we derive a Harnack type inequality along with a Liouville type theorem.

Weak solutions of Anisotropic (and crystalline) inverse mean curvature flow as limits of p-capacitary potentials

Tue, 02 Apr 2024 00:00:00 GMT

Esther Cabezas-Rivas, Salvador Moll, Marcos Solera — We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in ℝ^N with no ellip\-ticity nor smoothness requirements, in order to include the crystalline case) and on the initial data. By means of an approximation procedure introduced by Moser, our solutions are limits of anisotropic p-harmonic functions or p-capacitary functions (after a change of variable), and we get uniqueness both for the approximating solutions (i.e., uniqueness of p-capacitary functions) and the limiting ones. Our notion of weak solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV-functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. Moreover, by assuming the extra regularity given by an interior rolling ball condition (where a sliding Wulff shape plays the role of a ball), the solutions are shown to be continuous and satisfy Harnack inequalities. Finally, examples of explicit solutions are built.

Li-Yau Estimates for a Nonlinear Parabolic Equation on Finsler Manifolds

Thu, 14 Dec 2023 00:00:00 GMT

Bin Shen, Yuhan Zhu — In this paper, we explore the positive solutions to the Finslerian nonlinear equation ∂u/∂t = Δ^(∇u) u + aulog u + bu, which is related to Ricci solitons and serves as the Euler-Lagrange equation to the Finslerian log-energy functional. We then obtain the global gradient estimate of its positive solution on a compact Finsler metric measure space with the weighted Ricci curvature bounded below. Furthermore, using a new comparison theorem developed by the first author, we also establish a local gradient estimate on a non-compact forward complete Finsler metric measure spaces with the mixed weighted Ricci curvature bounded below, as well as finite bounds of misalignment and some non-Riemannian curvatures. Lastly, we prove the Harnack inequalities and a Liouville-type theorem of such solutions.

Parabolic frequency monotonicity for two nonlinear equations under Ricci flow

Mon, 23 Oct 2023 00:00:00 GMT

Chuanhuan Li, Yi Li, Kairui Xu, Jichun Zhu — In this paper, we consider the parabolic frequency for positive solutions of two nonlinear parabolic equations under the Ricci flow on closed manifolds. We obtain the monotonicity of parabolic frequency for the solution of two nonlinear parabolic equations with bounded Ricci curvature, then we apply the parabolic frequency monotonicity to get some integral type Harnack inequalities and we use -K1 instead of the lower bound 0 of Ricci curvature from Theorem 4.3 in 16, where K1 is any positive constant.

Finite time singularities of the Kähler-Ricci flow

Wed, 11 Oct 2023 00:00:00 GMT

Wangjian Jian, Jian Song, Gang Tian — We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano Kähler-Ricci flow to general finite time solutions of the Kähler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the Kähler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the Kähler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

Wed, 11 Oct 2023 00:00:00 GMT

Wangjian Jian, Jian Song, Gang Tian — In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new approach initiates the work in for general finite time solutions of the Kähler-Ricci flow.

Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow

Thu, 20 Jul 2023 00:00:00 GMT

Xiaolong Li, Hao-Yue Liu, Xin-An Ren — We prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Kähler-Ricci flow. As an application, we obtain a monotonicity formula.

A direct approach to sharp Li-Yau Estimates on closed manifolds with negative Ricci lower bound

Sat, 08 Jul 2023 00:00:00 GMT

Xingyu Song, Ling Wu, Meng Zhu — Recently, Qi S.Zhang [26] has derived a sharp Li-Yau estimate for positive solutions of the heat equation on closed Riemannian manifolds with the Ricci curvature bounded below by a negative constant. The proof is based on an integral iteration argument which utilizes Hamilton's gradient estimate, heat kernel Gaussian bounds and parabolic Harnack inequality. In this paper, we show that the sharp Li-Yau estimate can actually be obtained directly following the classical maximum principle argument, which simplifies the proof in [26]. In addition, we apply the same idea to the heat and conjugate heat equations under the Ricci flow and prove some Li-Yau type estimates with optimal coefficients.

Horizontal inverse mean curvature flow in the Heisenberg group

Tue, 27 Jun 2023 00:00:00 GMT

Jingshi Cui, Peibiao Zhao — Huisken and Ilmanen [J. Differential Geom., 2001] created the theory of weak solutions for inverse mean curvature flows (IMCF) of hypersurfaces on Riemannian manifolds, and proved successfully a Riemannian version of the Penrose inequality. The present paper investigates and constructs a sub-Riemannian version of the theory of weak solutions for inverse mean curvature flows of surfaces in the first Heisenberg group ℍ¹. The level set formulation of the IMCF in ℍ¹ is given by (0.1), where Ω⊂ ℍ¹ is an open set with smooth boundary, and Ω^(c) = ℍ¹∖ Ω= \ u ≤ 0\ is bounded. Let wₚ = exp ( (uₚ)/(1-p)) and wₚ satisfies (0.2). Following the argument by Moser, the key ingredient in proving the existence of weak solutions to (0.1) is to establish a uniform interior estimate for |∇₀ uₚ|. However, due to the lack of boundary continuity of |∇₀ uₚ| = (p-1)frac|∇₀ wₚ|wₚ ∈ C^(0,β)(Ω) (0< β<1, p>1) by Zhong and Mukherjee [Anal. PDE, 2021], the standard method in [R. Moser, J. Eur. Math. Soc., 2007] cannot be applied to obtain a uniform interior estimate for |∇₀ uₚ|. Fortunately, the present paper discovers two refined inequalities: Harnack inequality and Lipschitz estimate for wₚ, which allow one to obtain interior estimates for |∇₀ uₚ| independent of p. By further combining them with Arzelgraverm a-Ascoli theorem, the weak solution of (0.1) can then be generated as the limit of uₚ as p → 1, where wₚ = exp ( (uₚ)/(1-p)) and wₚ is of solutions to (0.2). As an important application of the IMCF in ℍ¹, a positive answer to an open problem posed in [F. Montefalcon, Ann. Mat. Pura Appl. (4), 2014]:Heintze-Karcher inequality in ℍ¹ is provided.

Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency

Fri, 16 Jun 2023 00:00:00 GMT

Xiaolong Li, Qi S. Zhang — In this paper we prove matrix Li-Yau-Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply such estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

Fri, 10 Mar 2023 00:00:00 GMT

Ali Taheri, Vahideh Vahidifar — This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to a nonlinear parabolic equation 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.