Noncompact Riemannian Manifolds Research Articles

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Let (M,g) be a complete non-compact Riemannian manifold together with a function eh, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of M to deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in M centered at a point o∈M. As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the distance function from o. The technique extends to non-compact submanifolds properly immersed in M under certain control on their weighted mean curvature.

Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature

In this paper we consider complete noncompact Riemannian manifolds (M, g) with nonnegative Ricci curvature and Euclidean volume growth, of dimension $$n \ge 3$$ . For every bounded open subset $$\Omega \subset M$$ with smooth boundary, we prove that $$\begin{aligned} \int \limits _{\partial \Omega } \left| \frac{\mathrm{H}}{n-1}\right| ^{n-1} \!\!\!\!\!{\mathrm{d}}\sigma \,\,\ge \,\,{\mathrm{AVR}}(g)\,\big |\mathbb {S}^{n-1}\big |, \end{aligned}$$ where $${\mathrm{H}}$$ is the mean curvature of $$\partial \Omega $$ and $${\mathrm{AVR}}(g)$$ is the asymptotic volume ratio of (M, g). Moreover, the equality holds true if and only if $$(M{{\setminus }}\Omega , g)$$ is isometric to a truncated cone over $$\partial \Omega $$ . An optimal version of Huisken’s Isoperimetric Inequality for 3-manifolds is obtained using this result. Finally, exploiting a natural extension of our techniques to the case of parabolic manifolds, we also deduce an enhanced version of Kasue’s non existence result for closed minimal hypersurfaces in manifolds with nonnegative Ricci curvature.

The Poisson equation on Riemannian manifolds with weighted Poincar\xe9 inequality at infinity

We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré inequalities outside compact sets. Our result applies to a large class of manifolds including, for instance, all non-parabolic manifolds with minimal positive Green’s function vanishing at infinity. On the source function, we assume a sharp pointwise decay depending on the weight appearing in the Poincaré inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold. In comparison with previous works, we can deal with a more general setting on the curvature bounds and without any spectral assumption.

The stochastic nonlinear Schrödinger equation in unbounded domains and non-compact manifolds

In this article, we construct a global martingale solution to a general nonlinear Schrödinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like $${{\mathbb {R}}^d}$$ , non-compact Riemannian manifolds, and unbounded domains in $${{\mathbb {R}}^d}$$ with different boundary conditions. The initial value belongs to the energy space $$H^1$$ and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood–Paley-decomposition. In the limit procedure, we employ tightness of the approximated solutions and Jakubowski’s extension of the Skorohod theorem to nonmetric spaces.

Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case

We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.

Nonnegative Scalar Curvature and Area Decreasing Maps

Let $\big(M,g^{TM}\big)$ be a noncompact complete spin Riemannian manifold of even dimension $n$, with $k^{TM}$ denote the associated scalar curvature. Let $f\colon M\rightarrow S^{n}(1)$ be a smooth area decreasing map, which is locally constant near infinity and of nonzero degree. We show that if $k^{TM}\geq n(n-1)$ on the support of ${\rm d}f$, then $ \inf \big(k^{TM}\big)< 0$. This answers a question of Gromov. We use a simple deformation of the Dirac operator to prove the result. The odd dimensional analogue is also presented.

Isometry-invariant solutions to a critical problem on non-compact Riemannian manifolds

We analyse an elliptic equation with critical growth set on a d-dimensional (d≥3) Hadamard manifold (M,g). By adopting a variational perspective, we prove the existence of non-zero non-negative solutions invariant under the action of a specific family of isometries. Our result remains valid when the original nonlinearity is singularly perturbed. Preserving the same variational approach, but considering other groups of isometries, we finally show that when M=Rd, d>3, and the nonlinearity is odd, there exist at least (−1)d+[d−32] pairs of sign-changing solutions.

Linear parabolic equations with strong boundary degeneration

As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in Euclidean spaces. The new feature of our result is the fact that—besides of obtaining an optimal solution theory—we consider the ‘natural’ case where the degeneration occurs only in the normal direction.

New Results About the Lambda Constant and Ground States of the 𝑊-Functional

Abstract In this paper, we study properties of the lambda constants and the existence of ground states of Perelman’s famous W-functional from a variational formulation. We have two kinds of results. One is about the estimation of the lambda constant of G. Perelman, and the other is about the existence of ground states of his W-functional, both on a complete non-compact Riemannian manifold ( M , g ) {(M,g)} . One consequence of our estimation is that, on an ALE (or asymptotic flat) manifold ( M , g ) {(M,g)} , if the scalar curvature s of ( M , g ) {(M,g)} is non-negative and has quadratical decay at infinity, then M is scalar flat, i.e., s = 0 {s=0} in M. We also introduce a new constant d ⁢ ( M , g ) {d(M,g)} . For the existence of the ground states, we use Lions’ concentration-compactness method.

The existence of ground state solution to elliptic equation with exponential growth on complete noncompact Riemannian manifold

In this paper, we consider the following elliptic problem: −divg(|∇gu|N−2∇gu)+V(x)|u|N−2u=f(x,u)a(x)in M,(Pa)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\mathtt{div}_{g}\\bigl( \\vert \\nabla_{g} u \\vert ^{N-2}\\nabla_{g} u \\bigr)+V(x) \\vert u \\vert ^{N-2}u = \\frac{f(x, u)}{a(x)}\\quad \\mbox{in } M, \\qquad (P_{a}) $$\\end{document} where (M, g) be a complete noncompact N-dimensional Riemannian manifold with negative curvature, Ngeq2, V is a continuous function satisfying V(x) geq V_{0 }> 0, a is a nonnegative function and f(x, t) has exponential growth with t in view of the Trudinger–Moser inequality. By proving some estimates together with the variational techniques, we get a ground state solution of (P_{a}). Moreover, we also get a nontrivial weak solution to the perturbation problem.

On the equivalence of heat kernels of second-order parabolic operators

Let P be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold M, and let V be a real valued function which belongs to the class of small perturbation potentials with respect to the heat kernel of P in M. We prove that under some further assumptions (satisfied by large classes of P and M) the positive minimal heat kernels of P - V and of P on M are equivalent. Moreover, the parabolic Martin boundary is stable under such perturbations, and the cones of all nonnegative solutions of the corresponding parabolic equations are affine homeomorphic.

Shnol-type theorem for the Agmon ground state

Let H be a Schrödinger operator defined on a noncompact Riemannian manifold \Omega , and let W\in L^\infty(\Omega;\mathbb{R}) . Suppose that the operator H+W is critical in \Omega , and let \varphi be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction of H satisfying |u|\leq \varphi in \Omega , then the corresponding eigenvalue is in the spectrum of H . The conclusion also holds true if for some K\Subset \Omega the operator H admits a positive solution in \tilde{\Omega}=\Omega\setminus K , and |u|\leq \psi in \tilde{\Omega} , where \psi is a positive solution of minimal growth in a neighborhood of infinity in \Omega . Under natural assumptions, this result holds true also in the context of infinite graphs, and Dirichlet forms.

The Kasparov product on submersions of open manifolds

We study the Kasparov product on (possibly non-compact and incomplete) Riemannian manifolds. Specifically, we show on a submersion of Riemannian manifolds that the tensor sum of a regular vertically elliptic operator on the total space and an elliptic operator on the base space represents the Kasparov product of the corresponding classes in KK-theory. This construction works in general for symmetric operators (i.e. without assuming self-adjointness), and extends known results for submersions with compact fibers. The assumption of regularity for the vertically elliptic operator is not always satisfied, but depends on the topology and geometry of the submersion, and we give explicit examples of non-regular operators. We apply our main result to obtain a factorization in unbounded KK-theory of the fundamental class of a Riemannian submersion, as a Kasparov product of the shriek map of the submersion and the fundamental class of the base manifold.

Superlinear elliptic inequalities on manifolds

Let M be a complete non-compact Riemannian manifold and let σ be a Radon measure on M. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequality−Δu≥σuqinM, where q>1. We obtain necessary and sufficient criteria for existence of positive solutions in terms of Green function of Δ. In particular, explicit necessary and sufficient conditions are given when M has nonnegative Ricci curvature everywhere in M, or more generally when Green's function satisfies the 3G-inequality.

Yamabe equation on some complete noncompact manifolds

In this paper, we consider the Yamabe equation on a complete noncompact Riemannian manifold and find some geometric conditions on the manifold such that the Yamabe problem admits a bounded positive solution.

On Jacobi-Type Vector Fields on Riemannian Manifolds

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds

In this note we prove a nonexistence result for proper biharmonic maps from complete non-compact Riemannian manifolds of dimension m=dimM≥3 with infinite volume that admit a Euclidean type Sobolev inequality into general Riemannian manifolds by assuming finiteness of ‖τ(ϕ)‖Lp(M),p>1 and smallness of ‖dϕ‖Lm(M). This is an improvement of a recent result of the first named author, where he assumed 2¡p¡m. As applications we also get several nonexistence results for proper biharmonic submersions from complete non-compact manifolds into general Riemannian manifolds.

F-biharmonic maps into general Riemannian manifolds

Abstract Let ψ:(M, g) → (N, h) be a map between Riemannian manifolds (M, g) and (N, h). We introduce the notion of the F-bienergy functional $$\begin{array}{} \displaystyle E_{F,2}(\psi)=\int\limits_{M}F\left(\frac{|\tau(\psi)|^{2}}{2}\right)\text{d}V_{g}, \end{array}$$ where F : [0, ∞) → [0, ∞) be C3 function such that F′ &gt; 0 on (0, ∞), τ(ψ) is the tension field of ψ. Critical points of τF,2 are called F-biharmonic maps. In this paper, we prove a nonexistence result for F-biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the Lp-norm (p &gt; 1) of the tension field is bounded and the m-energy of the maps is sufficiently small, then every F-biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].

On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifolds

On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifolds

A characterization related to Schrödinger equations on Riemannian manifolds

In this paper, we consider the following problem: [Formula: see text] where [Formula: see text] is a [Formula: see text]-dimensional ([Formula: see text], non-compact Riemannian manifold with asymptotically non-negative Ricci curvature, [Formula: see text] is a real parameter, [Formula: see text] is a positive coercive potential, [Formula: see text] is a bounded function and [Formula: see text] is a suitable nonlinearity. By using variational methods, we prove a characterization result for existence of solutions for [Formula: see text].