Noncompact Riemannian Manifolds Research Articles

Local Hamilton Type Gradient Estimates for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper, we present a new proof and attain some new local Hamilton type gradient estimates for positive solutions to the parabolic equation ut=ΔVup+aulogu+buon a complete noncompact Riemannian manifold with k-Bakry-Émery Ricci curvature bounded from below, where p, a and b are some given constants. As applications, related local Hamilton type gradient estimates, some parabolic type Liouville theorems and Harnack inequalities for porous media type equation and fast diffusion type equation are established. Some known results were generalized by our results.

Asymptotic equivalence of identification operators in geometric scattering theory

Given two measures \mu_{1} and \mu_{2} on a measurable space X such that d\mu_{2}=\rho_{1,2}\,d\mu_{1} for some bounded measurable function \rho_{1,2}:X\rightarrow (0,\infty) , there exist two natural identification operators J_{1,2},\tilde{J}_{1,2}:L^{2}(X,\mu_{1})\rightarrow L^{2}(X,\mu_{2}) , namely the unitary J_{1,2}\psi:=\psi/\sqrt{\rho_{1,2}} and the trivial \tilde{J}_{1,2}\psi:=\psi . Given self-adjoint semibounded operators H_{j} on L^{2}(X,\mu_{j}) , j=1,2 , we prove a natural criterion in a topologic setting for the equality of the two-Hilbert-space wave operators W_{\pm}(H_{2},H_{1};J_{1,2}) and W_{\pm}(H_{2},H_{1};\tilde{J}_{1,2}) , by showing that J_{1,2}-\tilde{J}_{1,2} are asymptotically H_{1} -equivalent in the sense of Kato. It turns out that this criterion is automatically satisfied in typical situations on noncompact Riemannian manifolds and weighted infinite graphs in which one has the existence of completeness W_{\pm}(H_{2},H_{1};\tilde{J}_{1,2}) (and thus a-posteriori of W_{\pm}(H_{2},H_{1};J_{1,2}) ).

Gradient estimates for Δpu + Δqu + a|u|s−1u + b|u|l−1u = 0 on a complete Riemannian manifold and applications

In this paper, we use the Nash–Moser iteration method to study the local and global behaviors of non-negative solutions to the nonlinear elliptic equation [Formula: see text] defined on a complete Riemannian manifold [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are constants and [Formula: see text], with [Formula: see text], is the usual [Formula: see text]-Laplace operator. Under some assumptions on [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], we derive gradient estimates and Liouville-type theorems for non-negative solutions to the above equation. In particular, we show that, if [Formula: see text] is a non-negative entire solution to [Formula: see text] ([Formula: see text]) on a complete non-compact Riemannian manifold [Formula: see text] with non-negative Ricci curvature and [Formula: see text], and [Formula: see text] where [Formula: see text] then [Formula: see text] is a trivial constant solution.

The quantization of Maxwell theory in the Cauchy radiation gauge: Hodge decomposition and Hadamard states

AbstractThe aim of this paper is to prove the existence of Hadamard states for the Maxwell equations on any globally hyperbolic spacetime. This will be achieved by introducing a new gauge fixing condition, the Cauchy radiation gauge, that will allow to suppress all the unphysical degrees of freedom. The key ingredient for achieving this gauge is a new Hodge decomposition for differential ‐forms in Sobolev spaces on complete (possibly noncompact) Riemannian manifolds.

Rigidities of isoperimetric inequality under nonnegative Ricci curvature

The sharp isoperimetric inequality for non-compact Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions culminated by Balogh and Kristály (2023) also covering metric-measure spaces satisfying the nonnegative Ricci curvature condition in the synthetic sense of Lott, Sturm and Villani. In sharp contrast with the compact case of positive Ricci curvature, for a large class of spaces including weighted Riemannian manifolds, no complete characterization of the equality cases is present in the literature. The scope of this paper is to settle this problem by proving, in the same generality as Balogh and Kristály (2023), that the equality in the isoperimetric inequality can be attained only by metric balls. Whenever this happens the space is forced, in a measure theoretic sense, to be a cone. Our result applies to different frameworks yielding as corollaries new rigidity results: it extends the rigidity results of Brendle (2023) for weighted Riemannian manifolds and the rigidity results of Antonelli et al. (2023) for general \mathsf{RCD} spaces. It also applies to the Euclidean setting by proving that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.

Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products

Abstract Our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions. Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker spacetimes such as the Einstein-de Sitter spacetime, are also given. Furthermore, we study the nonexistence of entire solutions to the mean curvature flow equation. Our approach is based on a suitable conformal change of metric jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al.

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

Revisiting Gradient Bach Solitons via Maximum Principles

Supposing that the Ricci curvature has an appropriate lower bound and applying suitable maximum principles, we establish triviality results which guarantee that a gradient Bach soliton must be trivial and Bach-flat. Our approach is based on three main cores: convergence to zero at infinity, polynomial volume growth (both related to complete noncompact Riemannian manifolds) and stochastic completeness.

The Log-Sobolev Inequality for a Submanifold in Manifolds With Asymptotic Non-Negative Intermediate Ricci Curvature

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong et al. (Acta Math. Sci. Ser. B (Engl. Ed.), 44(1):189–194 (2024)) which already generalizes Yi and Zheng (To appear in Chin. Ann. Math., (2023)) and Brendle (Comm. Pure Appl. Math. 75(3), 449–454 (2022)).

Symptotic behavior of solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds

The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on non-compact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary non-compact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.

Existence of solutions to modified nonlinear Schrödinger equations on non-compact Riemannian manifolds

We study the existence of nontrivial solutions to a class of modified nonlinear Schrödinger equation on a complete non-compact N-dimensional (N≥3) Riemannian manifold with asymptotically non-negative Ricci curvature. Using the critical point theory, in a general Sobolev space instead of an Orlicz one, we prove the existence of a nontrivial non-negative solution to a class of modified Schrödinger equation with coercive potential. We also estimate the decay rate of the solution and show that the solution is exponentially decay.

Asymptotic Behavior of Solutions of the Inhomogeneous Schrödinger Equation on Noncompact Riemannian Manifolds

Asymptotic Behavior of Solutions of the Inhomogeneous Schrödinger Equation on Noncompact Riemannian Manifolds

Evolution Flows on Non-compact Riemannian Manifolds: Stability, Periodicity, and Applications

Evolution Flows on Non-compact Riemannian Manifolds: Stability, Periodicity, and Applications

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document} initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds M with Ricci curvature bounded from below, positive injectivity radius and spectral gap.Our first main result states that, if L is the positive Laplace–Beltrami operator on M, then the Riesz–Hardy space HR1(M) is the isomorphic image of the Goldberg type space h1(M) via the map L1/2(I+L)−1/2, a fact that is false in Rn. Specifically, HR1(M) agrees with the Hardy type space X1/2(M) recently introduced by the first three authors; as a consequence, we prove that HR1(M) does not admit an atomic characterisation.Noninclusions are mostly proved in the special case where the manifold is a Damek–Ricci space S. Our second main result states that HR1(S), the heat Hardy space HH1(S) and the Poisson–Hardy space HP1(S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.

Global sharp gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Abstract In this paper, we employ the techniques in [C. Cavaterra, S. Dipierro, Z. Gao and E. Valdinoci, Global gradient estimates for a general type of nonlinear parabolic equations, J. Geom. Anal. 32 2022, 2, Paper No. 65] and the approach in [H. T. Dung and N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Amer. Math. Soc. 147 2019, 12, 5329–5338] to derive sharp gradient estimates for a positive solution to the heat equation u t = Δ ⁢ u + a ⁢ u ⁢ log ⁡ u u_{t}=\Delta u+au\log u in a complete noncompact Riemannian manifold (where a is a real constant). This is an extension of the gradient estimates of Dung and Dung.

The fractional porous medium equation on noncompact Riemannian manifolds

The fractional porous medium equation on noncompact Riemannian manifolds

Stabilization of the critical semilinear wave equation on non-compact Riemannian manifold

We study the stability of the defocusing critical semilinear wave equation with locally distributed damping in an exterior domain. Under some assumptions on the Riemannian metric, the geometric multiplier technique and the compactness-uniqueness arguments were used to prove the unique continuation, the observability inequality respectively. Then the energy decay of the nonlinear system was obtained.