Noncompact Manifolds Research Articles

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.

Differential Operators on Non-compact Harmonic Manifolds

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function. Furthermore, we show that the algebra of differential operators that commute with taking averages over geodesic spheres is generated by the Laplacian. As an application of this, we show that the heat-semi group is dense in the radial 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} space of a non-compact simply connected harmonic manifold. In the process, we provide a characterisation of the eigenfunctions of the Laplacian and therefore of the eigenfunctions of all differential operators commuting with taking spherical averages.

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.

An infinite–rank Lie algebra associated to SL(2,R) and SL(2,R)/U(1)

We construct a generalised notion of Kac-Moody algebras using smooth maps from the non-compact manifolds M= SL(2,R) and M= SL(2,R)/U(1) to a finite-dimensional simple Lie group G. This construction is achieved through two equivalent ways: by means of the Plancherel Theorem and by identifying a Hilbert basis within L2(M). We analyze the existence of central extensions and identify those in duality with Hermitean operators on M. By inspecting the Clebsch–Gordan coefficients of sl(2,R), we derive the Lie brackets characterising the corresponding generalised Kac-Moody algebras. The root structure of these algebras is identified, and it is shown that an infinite number of simultaneously commuting operators can be defined. Furthermore, we briefly touch upon applications of these algebras within the realm of supergravity, particularly in scenarios where the scalar fields coordinatize the non-compact manifold SL(2,R)/U(1).

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

The L2 Aeppli-Bott-Chern Hilbert complex

We analyse the L2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.

Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

In this paper, we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.

P-Parabolicity and a Generalized Bochner’s Method with Applications

The notions of function growth and sharp global integral estimates in p-harmonic geometry in Wei (Contemp Math 756:247–269, 2020) and Wei et al. (Sharp estimates on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}$$\\end{document}-harmonic functions with applications in biharmonic maps) naturally lead to a generalized uniformization theorem and a generalized Bochner’s method. These tools enable one to explore various geometric and variational problems in complete noncompact manifolds of arbitrary dimensions. In particular, we find the first set of nontrivial geometric quantities that are p-subharmonic and p-superharmonic functions (cf. Sect. 4.2). As further applications, we establish Liouville theorems for nonnegative A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {A}}$$\\end{document}-superharmonic functions (cf. Theorem 2.1) and for p-harmonic morphisms (cf. Theorem 4.3), Picard type theorems (cf. Sect. 4.3), existence theorems of harmonic maps (cf. Theorem 4.6), and a solution of the generalized Bernstein problem under p-parabolicity condition (without any volume growth condition, cf. Theorem 5.1) or under p-moderate volume growth (2.4) (cf. Corollary 5.1). These findings generalize and extend the work of Schoen–Simon–Yau under volume growth condition (0.1) which is due to (i)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {i}})$$\\end{document} Corollary 2.1(iv), which states that a manifold with p-moderate volume growth (2.4) must be p-parabolic (generalizing a result of Cheng and Yau (Commun Pure Appl 28(3):333–354, 1975) for the case p=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=2$$\\end{document}, F(r)≡1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F(r)\\equiv 1$$\\end{document}), (ii)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {ii}})$$\\end{document} Example 2.1 of a p-parabolic manifold with exponential volume growth, and (iii)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({\ ext {iii}})$$\\end{document} volume growth condition (0.1) ⟹(implies)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overset{(\ ext {implies})}{\\Longrightarrow }$$\\end{document}p-moderate volume growth (2.4). Furthermore, we offer applications to p-harmonic maps and stable p-harmonic maps. Notably, these outcomes address an intriguing question posed by Kobayashi if p-subharmonic functions are valuable tools for studying p-harmonic maps or related topics (cf. in: Kobayashi, Midwest Geometry Conference, University of Oklahoma, Oklahoma, 2006. Private Communication, 1996), and answer in the affirmative. Other applications of the notions and estimates to biharmonic maps, isometric immersions, holomorphic functions on Kähler manifolds, generalized harmonic forms, and Yang–Mills fields on complete noncompact manifolds can be found in Wei et al. (Sharp estimates on A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal A$$\\end{document}-harmonic functions with applications in biharmonic maps), Chen and Wei (Glasg Math J 51(3):579–592, 2009), Wei (Bull Transilv Univ Brasov Ser III 1(50):415–453, 2008), Wei (Growth estimates for generalized harmonic forms on noncompact manifolds with geometric applications, Contemp Math 756 American Mathematical Society, Providence, 247–269, 2020), Wei (Sci China Math 64(7):1649–1702, 2021) and Wei (Isolation phenomena for Yang–Mills fields on complete manifolds). We generate the work of Schoen–Simon–Yau under p-parabolic condition, in which the result can be used in other types of new manifolds we found, by an extrinsic average variational method we proposed (cf. Wei in Rom J Math Comput Sci 13(2):100–124, 2023; in: Wei, Connecting Poincare inequality with Sobolev inequality on Riemannian manifolds, Int Electron J Geom 17(1):290–305, 2024).

$L^2$-estimates for the Dirac-Dolbeault operator and Bergman kernel asymptotics on some classes of non-compact complex manifolds

$L^2$-estimates for the Dirac-Dolbeault operator and Bergman kernel asymptotics on some classes of non-compact complex manifolds

On ends of finite-volume noncompact manifolds of nonpositive curvature

On ends of finite-volume noncompact manifolds of nonpositive curvature

Canonical Metrics on Twisted Quiver Bundles over a Class of Non-Compact Gauduchon Manifold

The aim of this paper is to prove a theorem for holomorphic twisted quiver bundles over a special non-compact Gauduchon manifold, connecting the existence of (σ,τ)-Hermite–Yang–Mills metric in differential geometry and the analytic (σ,τ)-stability in algebraic geometry. The proof of the theorem relies on the flow method and the Uhlenbeck–Yau’s continuity method.

Asymptotic normality of divisors of random holomorphic sections on non-compact complex manifolds

We prove a central limit theorem for divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

Geometric conditions to obtain Anosov geodesic flow for non-compact manifolds

Let ( M , g ) be a complete Riemannian manifold without focal points and curvature bounded below. We prove that when the average of the sectional curvature in tangent planes along geodesics is negative and uniformly away from zero, then the geodesic flow is of Anosov type. We use this result to construct a non-compact manifold with non-positive curvature, without compact quotient and geodesic flow of Anosov type.

Introduction to mean dimension

The purpose of this survey is to explain basic ideas of mean dimension theory. Mean dimension represents a number of parameters per unit time for describing a dynamical system. It was introduced by Gromov in 1999. His motivation is to propose a new approach to geometric analysis over noncompact manifolds. Rather independently of this original motivation, Lindenstrauss and Weiss found applications of mean dimension to classical problems in topological dynamics. In this survey we start from the viewpoint of Lindenstrauss–Weiss and first review the applications to topological dynamics. Next we explain the relation to geometric analysis, in particular holomorphic curve theory. Mean dimension is connected to rate distortion theory, which is a branch of information theory studying the trade-off between distortion and compression rate in lossy data compression. In the last three sections we review a variational principle between mean dimension and rate distortion function and then discuss its possible application to holomorphic curve theory.

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.

Connecting Poincaré Inequality with Sobolev Inequalities on Riemannian Manifolds

We connect the Poincaré inequality with the Sobolev inequality on Riemannian manifold in a family of integral inequalities $(1.5)$. For these continuum of inequalities, we obtain topological structure theorems of manifolds generalizing previous unification theorems in both intrinsic and extrinsic settings ([33]). Manifolds which admit any of these integral inequalities are nonparabolic, affect topology, geometry, analysis, and admit nonconstant bounded harmonic functions of finite energy. As a consequence, we have proven a Conjecture of Schoen-Yau ([27, p.74]) to be true in dimension two with hypotheses weaker than that used in [1] and [33]$($ which were weaker than the hypotheses set in the conjecture, $($ cf. Remark 1.5$)$. In the same philosophy and spirit as in ([31]), we prove that if $M$ is a complete $n$-manifold, satisfying $\operatorname{(i)}$ the volume growth condition $(1.1)$, $\operatorname{(ii)}$ Liouville Theorem for harmonic functions, and either $\operatorname{(v)}$ a generalized Poincaré- Sobolev inequality $(1.5)$, or $\operatorname{(vi)}$ a general integral inequality $(1.6)$, and Liouville Theorem for harmonic map $u : M \to K$ with $\operatorname{Sec}^K \le 0$, then $(1)$ $M$ has only one end and $(2)$ there is no nontrivial homomorphism from fundamental group $\pi_1(\partial D)$ into $\pi_1 (K)$ as stated in Theorem $1.5$. Some applications in geometry $(\S 3)$, geometric analysis $(\S 4)$, nonlinear partial differential systems $(\S 5)$, integral inequalities on complete noncompact manifolds $(\S 6)$ are made $($cf. e.g., Theorems $3.1$, $4.1$, $5.1$, and $6.1)$. Whereas we made the first study in ([29, 32]) on how the existence of an essential positive supersolution of a second order partial differential systems $Q(u)=0$ on a Riemannian manifold $M$, (by which we mean a $C^2$ function $v \ge 0$ on $M$ that is positive almost everywhere on $M$, and that satisfies $Q(v)=\operatorname{div}(A(x,v,\nabla v)\nabla v)+b(x,v,\nabla v)v\leq 0\quad (5.1)\, $) affects topology, geometry, analysis and variational problems on the manifold $M$. Whereas we generate the work in [35], under $p$-parabolic stable condition without assuming the $p$-th volume growth condition $\lim _{r \to \infty} r^{-p}\operatorname{Vol}(B_r) =0$. The techniques, concepts, and results employed in this paper can be combined with those of essential positive supersolutions of degenerate nonlinear partial differential systems $($cf. for example, Theorems 5.1 - 5.5, 6.1, etc.$)\, $ generalizing previous work in [32, 4.11], which in term recaptures the work of Schoen-Simon-Yau ([25, Theorem 2]). The combined techniques, concepts and method of [32] and [35] can also be used in other new manifolds we found by an extrinsic average variational method ([34]).

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

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)).