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We study strong half-space theorems for the classes of complete 1-surfaces with bounded curvature, parabolic 1-surfaces, and stochastically complete H-surfaces with H < 1 immersed in the hyperbolic space H 3 . As a by-product of the techniques we obtain a Maximum Principle at Infinity for 1-surfaces in H 3 . We also address the intersection problem for 1-surfaces immersed in a complete Riemannian three-manifold P with Ricci curvature bounded from below by - 2 . We establish a splitting result provided the distance between the 1-surfaces is realized and Ric P ≥ - 2 , and a Frankel’s type theorem for 1-surfaces with bounded curvature immersed in P when Ric P > - 2 . [ABSTRACT FROM AUTHOR]

In this short note we show that Doyle-Grigor'yan criterion for non-parabolicity is not necessary in dimension greater than or equal to four. This gives an negative answer to Problem # 1 of Grigor'yan [Bull. Amer. Math. Soc. (N.S) 36 (1999). pp. 135–249] in this dimensional range. [ABSTRACT FROM AUTHOR]

We present a clear and practical way to characterize the parabolicity of a complete immersed surface that is invariant with respect to a Killing vector field of the ambient space. [ABSTRACT FROM AUTHOR]

Large scale properties of Riemannian manifolds, in particular, those properties preserved by quasi-isometries, can be studied using discrete structures which approximate the manifolds. In a sequence of papers, M. Kanai proved that, under mild conditions, many properties are preserved by a certain (quasi-isometric) graph approximation of a manifold. One of these properties is p-parabolicity. A manifold M (respectively, a graph G) is said to be p-parabolic if all positive p-superharmonic functions on M (resp. G) are constant. This is equivalent to not having p-Green's function (i.e. a positive fundamental solution of the p-Laplace-Beltrami operator). Herein we study directly the p-parabolicity on graphs. We obtain some characterizations in terms of graph decompositions. Also, we give necessary and sufficient conditions for a uniform hyperbolic graph to be p-parabolic in terms of its boundary at infinity. Finally, we prove that if a uniform hyperbolic graph satisfies the (Cheeger) isoperimetric inequality, then it is non-p-parabolic for every 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. We focus on the setting of geometrically finite Kleinian groups with parabolic elements and parabolic rational maps. In this context an especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two families. [ABSTRACT FROM AUTHOR]

In this article we deal with gradient h-almost Yamabe solitons introduced by Zeng (J Math Study 54(4):371–386 2021). In this setting we prove that under some conditions in the potential function, Ricci curvature or parabolicity of M, then the scalar curvature is a constant. Also proved that under some integral conditions, a gradient h-Yamabe soliton most be scalar curvature vanished. [ABSTRACT FROM AUTHOR]

In this paper, we deal with complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Einstein spacetime obeying standard curvature constraints. In this setting, we establish a parabolicity criterion related to a suitable Cheng–Yau modified operator and we apply it to obtain sufficient conditions which guarantee that such a spacelike hypersurface must be either totally umbilical or isometric to an isoparametric spacelike hypersurface with two distinct principal curvatures one of which is simple. Furthermore, an application to the de Sitter space is also given. [ABSTRACT FROM AUTHOR]

This work aims to provide some triviality results for complete Cotton solitons under different assumptions like non-positive Ricci curvature, convergence to zero at infinity, polynomial or volume growth, and stochastic completeness. Finally, as an application, we classify the Cotton solitons with additional curative condition to be isometric to a space form of constant sectional curvature or to a Riemannian product M 2 (c) × N 1 , where N 1 = R 1 or S 1 for some c. [ABSTRACT FROM AUTHOR]

This paper investigates the initial-boundary value problem for an integro-differential equation of parabolic type with a differentiation operator with respect to multidimensional time. A fundamental solution is constructed and some of its properties associated with multiperiodicity in time and diagonal periodicity in space variables are discovered. An estimate is obtained for the fundamental solution of an integro-differential equation. The existence and uniqueness of multiperiodic solutions of the boundary value problem with respect to all arguments are established and some basic properties of the desired solutions are found. [ABSTRACT FROM AUTHOR]

In this paper, we deal with complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector and flat normal bundle in an Einstein manifold Ɛn+pp p of index p. Under some curvature constraints on the ambient space, we establish an L-parabolicity criterion related to a suitable modified Cheng-Yau's operator L and we apply it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be an isoparametric hypersurface of Ɛn+11 1 with two distinct principal curvatures one of which is simple. [ABSTRACT FROM AUTHOR]

We give characterizations for the parabolicity of regular trees. [ABSTRACT FROM AUTHOR]

Kanai proved that quasi-isometries between Riemannian manifolds with bounded geometry preserve many global properties, including the existence of Green's function, i.e., non-parabolicity. However, Kanai's hypotheses are too restrictive. Herein we prove the stability of p-parabolicity (with 1 < p < ∞ ) by quasi-isometries between Riemannian manifolds under weaker assumptions. Also, we obtain some results on the p-parabolicity of graphs and trees; in particular, we characterize p-parabolicity for a large class of trees. [ABSTRACT FROM AUTHOR]

We study some potential theoretic properties of homothetic solitons Σ n of the MCF and the IMCF. Using the analysis of the extrinsic distance function defined on these submanifolds in R n + m , we observe similarities and differences in the geometry of solitons in both flows. In particular, we show that parabolic MCF-solitons Σ n with n > 2 are self-shrinkers and that parabolic IMCF-solitons of any dimension are self-expanders. We have studied too the geometric behavior of parabolic MCF and IMCF-solitons confined in a ball, the behavior of the mean exit time function for the Brownian motion defined on Σ as well as a classification of properly immersed MCF-self-shrinkers with bounded second fundamental form, following the lines of Cao and Li (Calc Var 46:879–889, 2013). [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a notion of mean curvature flow soliton in Riemannian ambient spaces general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces. As expected, our definition is motivated by the self-similarity of certain special solutions of the mean curvature flow with respect to the flow generated by a distinguished vector field on the target manifold. Our approach allows us to identify some natural geometric quantities that satisfy elliptic equations or differential inequalities in a simple and manageable form for which the machinery of weak maximum principles is valid. The latter is one of the main tools we apply to derive several new characterizations and rigidity results for mean curvature flow solitons that extend to our much more general setting known properties, for instance, in Euclidean space. [ABSTRACT FROM AUTHOR]

We study space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products , where F is a (non-compact) complete Riemannian manifold whose universal covering is parabolic. In particular, we provide several rigidity results under appropriate mathematical and physical assumptions. As an application, several Calabi–Bernstein-type results are obtained which widely extend the previous ones in this setting. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate complete generic shrinking Ricci solitons with pointwise pinched curvature and prove two classification results for such manifolds. In particular, we show that any n -dimensional generic shrinking Ricci soliton ( M n , g , X ) is isometric to a finite quotient of R n or S n under some pointwise pinched curvature condition. The arguments mainly rely on the Δ X -type parabolicity of ( M n , g , X ) and algebraic curvature estimates. [ABSTRACT FROM AUTHOR]

We provide natural geometric assumption for a minimal graph in a Riemannian warped product to be parabolic. As a consequence, we deal with several Bernstein-type problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the 3-dimensional Euclidean space or in a 3-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic 3-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution. [ABSTRACT FROM AUTHOR]

Motivated by recent interest in global analysis of singular manifolds, we establish the essential self-adjointness of a Laplacian, a Liouville property of subharmonic functions, conservativeness and parabolicity of an incomplete manifold. These results are applicable for manifolds with fractal Cauchy boundary.

Let Mm, with m ≥ 3, be an m-dimensional complete non-compact manifold isometrically immersed in a Hadamard manifold M. Assume that the mean curvature vector has finite Lp-norm, for some 2 ≤ p ≤ m. We prove that each end of M must either have finite volume or be non-parabolic. [ABSTRACT FROM AUTHOR]

In this paper, we study complete noncompact pseudohermitian manifolds with positive spectrum of the sub-Laplacian. We prove splitting-type theorems for a class of complete noncompact pseudohermitian manifolds with vanishing torsion whose spectrum of the sub-Laplacian has an optimal positive lower bound. These can be viewed as the CR analogue of theorems of Li-Wang and the equality case of a theorem of Cheng. [ABSTRACT FROM AUTHOR]

Abstract: Complete maximal hypersurfaces in Generalized Robertson–Walker spacetimes whose fiber has a parabolic universal Riemannian covering are studied. Under natural boundedness assumptions, a spacelike hypersurface is shown to be parabolic when it is endowed with a conformal metric. This fact allows us to get, for complete maximal hypersurfaces, several uniqueness results. As an application, all the entire solutions to the maximal hypersurface equation under certain constrains are obtained, solving new Calabi–Bernstein problems. [Copyright &y& Elsevier]

Abstract: In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a large family of uniformly elliptic EDP’s, so solving the Calabi–Bernstein problem in a wide family of spacetimes. [Copyright &y& Elsevier]

Abstract: We present and prove new characterizations of parabolicity and stochastic completeness for a general weighted manifold M as well as the uniqueness of the Markov extensions of the Laplacian in terms of Greenʼs formula. Moreover, we study the relationship between those properties and the singularity of M in terms of a fractal dimension and capacity. [Copyright &y& Elsevier]

Set in the Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, which in some sense are modeled after the p- Laplacian with potential. In particular, we discuss the equivalence between the Liouville property and the Khas'minskii condition, i.e. the existence of an exhaustion function which is also a supersolution for the operator outside a compact set. This generalizes a previous result obtained by one of the authors. [ABSTRACT FROM AUTHOR]

We establish parabolicity and quadratic area growth for minimal surfaces-with-boundary contained in regions of ℝ which are within a sub-logarithmic factor of the exterior of a cone. Unlike previous work showing that these two properties hold for minimal surfaces-with-boundary contained between two catenoids, we do not make use of universal superharmonic functions. Instead, we use stochastic methods, which have the additional feature of giving a type of parabolicity in a more general context than Brownian motion on a minimal surface. [ABSTRACT FROM AUTHOR]

Based on ideas of Pigolla and Setti (), we prove that isometrically immersed submanifolds with bounded mean curvature into Cartan-Hadamard manifolds are Feller. We consider Riemannian submersions π : M → N with compact minimal fibers and prove that the total space M is Feller, parabolic or stochastically complete if, and only if, the base manifold N is, respectively, Feller, parabolic or stochastically complete. [ABSTRACT FROM AUTHOR]

We prove the following version of Poincaré duality for reduced L-cohomology: For any 1 < q, p < ∞, the L-cohomology of a Riemannian manifold is in duality with the interior L-cohomology for 1/ p + 1/ p′ = 1/ q + 1/ q′ = 1. [ABSTRACT FROM AUTHOR]

We consider the invertibility of parabolic pseudodifferential operators in exponential weighted Sobolev spaces. We suppose that the symbol a of the operator Op(a) is analytically extended with respect to the impulse variable in an unbounded tube domain ℝn + iD and satisfies conditions of uniform parabolicity. We prove that under these conditions the pseudodifferential operator Op(a) is invertible in admissible weighted Sobolev spaces with weights connected with the domain D. As an application we obtain exponential estimates of solutions (including estimates of the fundamental solution) for parabolic differential operators. [ABSTRACT FROM AUTHOR]

We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $${M^2 \times \mathbb {R}_1}$$ , where M is a connected Riemannian surface with non-negative Gaussian curvature and $${M^2 \times \mathbb {R}_1}$$ is endowed with the Lorentzian product metric $${{\langle , \rangle}={\langle , \rangle}_M-dt^2}$$ . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain $${\Omega \subseteq M}$$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal graphs in $${M^2 \times \mathbb {R}_1}$$ . [ABSTRACT FROM AUTHOR]

We obtain upper bounds for the isoperimetric quotients of extrinsic balls of submanifolds in ambient spaces which have a lower bound on their radial sectional curvatures. The submanifolds are themselves only assumed to have lower bounds on the radial part of the mean curvature vector field and on the radial part of the intrinsic unit normals at the boundaries of the extrinsic spheres, respectively. In the same vein we also establish lower bounds on the mean exit time for Brownian motions in the extrinsic balls, i.e. lower bounds for the time it takes (on average) for Brownian particles to diffuse within the extrinsic ball from a given starting point before they hit the boundary of the extrinsic ball. In those cases, where we may extend our analysis to hold all the way to infinity, we apply a capacity comparison technique to obtain a sufficient condition for the submanifolds to be parabolic, i.e. a condition which will guarantee that any Brownian particle, which is free to move around in the whole submanifold, is bound to eventually revisit any given neighborhood of its starting point with probability 1. The results of this paper are in a rough sense dual to similar results obtained previously by the present authors in complementary settings where we assume that the curvatures are bounded from above. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori–Yau maximum principle due to Pigola, Rigoli, Setti (2005) . [Copyright &y& Elsevier]

Abstract: In this paper we establish new Calabi–Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form , where is a connected Riemannian surface and is endowed with the Lorentzian metric . In particular, we prove that when  is a Riemannian surface with non-negative Gaussian curvature , any complete maximal surface in  must be totally geodesic. Besides, if  is non-flat we conclude that it must be a slice , (here by complete it is meant, as usual, that the induced Riemannian metric on the maximal surface from the ambient Lorentzian metric is complete). We prove that the same happens if the maximal surface is complete with respect to the metric induced from the Riemannian product . This allows us to give also a non-parametric version of the Calabi–Bernstein theorem for entire maximal graphs in , under the same assumptions on . Moreover, we also construct counterexamples which show that our Calabi–Bernstein results are no longer true without the hypothesis . These examples are constructed via a duality result between minimal and maximal graphs. [Copyright &y& Elsevier]

In this article we have proved that a gradient Yamabe soliton satisfying some additional conditions must be of constant scalar curvature. Later, we have showed that in a gradient expanding Yamabe soliton with non-negative scalar curvature if the potential function satisfies an integral condition then it is isometric to the Euclidean space R n , and for steady case with the same condition the potential function becomes harmonic. Also we have proved that, in a compact gradient Yamabe soliton, the potential function agrees with the Hodge-de Rham potential up to a constant. [ABSTRACT FROM AUTHOR]