Noncompact Manifolds Research Articles

CR Gradient Estimate for the Positive Eigenfunction of Witten Sub-Laplacian

In this paper, under m-Bakry–Emery (or $$\infty $$ -Bakry–Emery) pseudohermitian Ricci curvature condition, we derive subgradient estimate for positive solutions of $$\Delta _{H,\phi } u(x)=-\lambda u(x)$$ on a weighted complete noncompact pseudohermitian manifold which satisfies the sub-Laplacian comparison property.

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.

CR Yamabe constant, CR Yamabe flow and its soliton

On a compact strictly pseudoconvex CR manifold (M,θ), we consider the CR Yamabe constant of its infinite conformal covering. By using the maximum principles, we then prove a uniqueness theorem for the CR Yamabe flow on a complete noncompact CR manifold. Finally we obtain some properties of the CR Yamabe soliton on complete noncompact CR manifolds.

Cohomology Dimension Growth for Nakano q-Semipositive Line Bundles

We study the cohomology with high tensor powers of Nakano $q$-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the order of estimates are optimal. Besides, estimates for the modified Dirac operator on Nakano $q$-positive line bundle on almost complex manifolds are given.

Chern–Ricci flows on noncompact complex manifolds

In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that any complete noncollapsed Kahler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kahler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume. Combining this result with the result of Chau-Tam, we give another proof that a complete noncompact Kahler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to the complex Euclidean space. This last result has already been proved by Gang Liu recently using other methods. This last result is partial confirmation of a uniformization conjecture of Yau.

Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold

In this paper, we study the minimization of λ1(Ω), the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets Ω of fixed volume in a Riemannian manifold (M,g). In the Euclidean setting (when (M,g)=(Rn,e)), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds (M,g). In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.

Rabinowitz Floer Homology for Tentacular Hamiltonians

Abstract This paper extends the definition of Rabinowitz Floer homology to non-compact hypersurfaces. We present a general framework for the construction of Rabinowitz Floer homology in the non-compact setting under suitable compactness assumptions on the periodic orbits and the moduli spaces of Floer trajectories. We introduce a class of hypersurfaces arising as the level sets of specific Hamiltonians: strongly tentacular Hamiltonians for which the compactness conditions are satisfied, cf. [ 21], thus enabling us to define the Rabinowitz Floer homology for this class. Rabinowitz Floer homology in turn serves as a tool to address the Weinstein conjecture and establish existence of closed characteristics for non-compact contact manifolds.

CR Nirenberg problem and zero Wester scalar curvature

In this paper, we study the prescribing Webster scalar curvature problem on strictly pseudoconvex CR manifolds of real dimension $$2n+1$$ . First, we study the CR Nirenberg problem and prove some existence results. Second, we provide the existence and a positive lower bound for a solution of the CR Yamabe problem with zero Webster scalar curvature on noncompact complete manifolds.

New features of the first eigenvalue on negatively curved spaces

Abstract The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model n-dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016) and Savo (2008); in odd dimensions, such eigenvalues appear as roots of an inductively constructed transcendental equation. We then give a synthetic proof of Cheng’s sharp eigenvalue comparison theorem in metric measure spaces satisfying a Bishop–Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.

On the κ-solutions of the Ricci flow on noncompact 3-manifolds

In this paper we prove that there is no κ-solution of Ricci flow on 3-dimensional noncompact manifold with strictly positive sectional curvature and blow up at some finite time T satisfying ∫0TT−tR(p0,t)dt<∞ for some point p0. This partially confirms a conjecture of Perelman.

Fourier transform, quantum mechanics and quantum field theory on the manifold of general relativity

A proof is given for the Fourier transform for functions in a quantum mechanical Hilbert space on a non-compact manifold in general relativity. In the (configuration space) Newton–Wigner representation, we discuss the spectral decomposition of the canonical operators and give a proof of the Parseval–Plancherel relation and the Born rule for linear superposition. We then discuss the representations of pure quantum states and their dual vectors and construct the Fock space and the associated quantum field theory for Bose–Einstein and Fermi–Dirac statistics.

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.

The Equivariant Coarse Novikov Conjecture and Coarse Embedding

The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group $\Gamma$ acts on a bounded geometric space $X$ properly, isometrically, and with bounded distortion, $X/\Gamma$ and $\Gamma$ admit coarse embeddings into Hilbert space, then the $\Gamma$-equivariant coarse Novikov conjecture holds for $X$. Here bounded distortion means that for any $\gamma\in\Gamma$, $\sup_{x\in Y} d(\gamma x,x)<\infty$, where $Y$ is a fundamental domain of the $\Gamma$-action on $X$.

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.

Image Set Classification Using a Distance-Based Kernel Over Affine Grassmann Manifold.

Modeling image sets or videos as linear subspaces is quite popular for classification problems in machine learning. However, affine subspace modeling has not been explored much. In this article, we address the image sets classification problem by modeling them as affine subspaces. Affine subspaces are linear subspaces shifted from origin by an offset. The collection of the same dimensional affine subspaces of [Formula: see text] is known as affine Grassmann manifold (AGM) or affine Grassmannian that is a smooth and noncompact manifold. The non-Euclidean geometry of AGM and the nonunique representation of an affine subspace in AGM make the classification task in AGM difficult. In this article, we propose a novel affine subspace-based kernel that maps the points in AGM to a finite-dimensional Hilbert space. For this, we embed the AGM in a higher dimensional Grassmann manifold (GM) by embedding the offset vector in the Stiefel coordinates. The projection distance between two points in AGM is the measure of similarity obtained by the kernel function. The obtained kernel-gram matrix is further diagonalized to generate low-dimensional features in the Euclidean space corresponding to the points in AGM. Distance-preserving constraint along with sparsity constraint is used for minimum residual error classification by keeping the locally Euclidean structure of AGM in mind. Experimentation performed over four data sets for gait, object, hand, and body gesture recognition shows promising results compared with state-of-the-art techniques.

Time analyticity for the heat equation and Navier-Stokes equations

We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order 2 on M. Here M is Rd or a complete noncompact manifold with Ricci curvature bounded from below by a constant. An implication is a sharp solvability condition for the Cauchy problem of the backward heat equation, which is a well known ill-posed problem. Another implication is a sharp criteria for time analyticity of solutions down to the initial time. The other pertains bounded mild solutions of the incompressible Navier-Stokes equations in the whole space.There are many long established analyticity results for the Navier-Stokes equations. See for example [17] and [10] for spatial and time analyticity in certain integral sense, [4] for pointwise space-time analyticity of 3 dimensional solutions to the Cauchy problem, and also the pointwise time analyticity results of [22] and [12] under zero boundary condition. Our result seems to be the first general pointwise time analyticity result for the Cauchy problem for all dimensions, whose proof involves only real variable method. The proof involves a method of algebraically manipulating the integral kernel, which appears applicable to other evolution equations.

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.