Noncompact Manifolds Research Articles

Sobolev functions without compactly supported approximations

A basilar property and a useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}(\R^n)$ (i.e. the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On Riemannian manifolds, it is well known that the same property remains valid under suitable geometric assumptions. However, on a complete non-compact manifold it can fail to be true in general, as we prove in this paper. This settles an open problem raised for instance by E. Hebey [\textit{Nonlinear analysis on manifolds: Sobolev spaces and inequalities}, Courant Lecture Notes in Mathematics, vol. 5, 1999, pp. 48-49].

Gap type results for spacelike submanifolds with parallel mean curvature vector

We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $\mathbb L_q^{n+p}(c)$ of index $1\leq q\leq p$ and constant sectional curvature $c\in \{-1,0,1\}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650).

Witten deformation on non-compact manifolds: heat kernel expansion and local index theorem

Asymptotic expansions of heat kernels and heat traces of Schrödinger operators on non-compact spaces are rarely explored, and even for cases as simple as {mathbb {C}}^n with (quasi-homogeneous) polynomials potentials, it’s already very complicated. Motivated by path integral formulation of the heat kernel, we introduced a parabolic distance, which also appeared in Li–Yau’s famous work on parabolic Harnack estimate. With the help of the parabolic distance, we derive a pointwise asymptotic expansion of the heat kernel for the Witten Laplacian with strong remainder estimate. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time t, and obtain a local index theorem. This is the second of our papers in understanding Landau–Ginzburg B-models on nontrivial spaces, and in subsequent work, we will develop the Ray–Singer torsion for Witten deformation in the non-compact setting.

Comparison Results for Poisson Equation with Mixed Boundary Condition on Manifolds

In this article, we establish a $$L^1$$ estimate for solutions to Poisson equation with mixed boundary condition, on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. On Riemann surfaces we obtain a Talenti-type comparison. Our results generalize main theorems in Alvino et al. (J Math Pures Appl 9(152):251–261, 2021) to Riemannian setting, and Chen–Li’s result (Talenti’s comparison theorem for poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature, 2021. arXiv:2104.05568 ) to the case of variable Robin parameter.

Non-degeneracy of Cohomological Traces for General Landau–Ginzburg Models

We prove non-degeneracy of the cohomological bulk and boundary traces for general open-closed Landau-Ginzburg models associated to a pair $(X,W)$, where $X$ is a non-compact complex manifold with trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$, assuming only that the critical locus of $W$ is compact (but may not consist of isolated points). These results can be viewed as certain "deformed" versions of Serre duality. The first amounts to a duality property for the hypercohomology of the sheaf Koszul complex of $W$, while the second is equivalent with the statement that a certain power of the shift functor is a Serre functor on the even subcategory of the $\mathbb{Z}_2$-graded category of topological D-branes of such models.

Hölder regularity and Liouville properties for nonlinear elliptic inequalities with power-growth gradient terms

This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set$\Omega$. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.

On the Generalized Geroch Conjecture for Complete Spin Manifolds

Let $W$ be a closed area enlargeable manifold in the sense of Gromov-Lawson and $M$ be a noncompact spin manifold, we show that the connected sum $M\# W$ admits no complete metric of positive scalar curvature. When $W=T^n$, this provides a positive answer to the generalized Geroch conjecture in the spin setting.

Nonexistence and umbilicity of spacelike submanifolds with second fundamental form locally timelike

Under the assumption that the second fundamental form is locally timelike, we establish new nonexistence and umbilicity results concerning n-dimensional spacelike submanifolds immersed with parallel mean curvature vector in the (n+p)-dimensional de Sitter space mathbb {S}^{n+p}_q of index q, such that 1le qle p. Our approach is based on a Simon’s type inequality involving the norm of the total umbilicity tensor, obtained by Mariano in [17], jointly with suitable maximum principles due to Alías, Caminha and do Nascimento [6, 7] for complete noncompact Riemannian manifolds and a weak version of Omori–Yau’s maximum principle for stochastically complete Riemanian manifolds proved by Pigola, Rigoli and Setti [20, 21].

Dirac operators on foliations with invariant transverse measures

We extend the groundbreaking results of Gromov and Lawson, [13], to Dirac operators defined along the leaves of foliations of non-compact complete Riemannian manifolds which admit invariant transverse measures. We prove a relative measured index theorem for pairs of such manifolds, foliations and operators, which are identified off compact subsets of the manifolds. We assume that the spectral projections of the leafwise operators for some interval [0,ϵ], ϵ>0, have finite dimensional images when paired with the invariant transverse measures. As a prime example, we show that if the zeroth order operators in the associated Bochner Identities are uniformly positive off compact subsets of the manifolds, then they satisfy the hypotheses of our relative measured index theorem. Using these results, we show that for a large collection of spin foliations, the space of positive scalar curvature metrics on each foliation has infinitely many path connected components.

Sobolev-Gaffney type inequalities for differential forms on sub-Riemannian contact manifolds with bounded geometry

Abstract In this article, we establish a Gaffney type inequality, in W ℓ , p {W}^{\ell ,p} -Sobolev spaces, for differential forms on sub-Riemannian contact manifolds without boundary, having bounded geometry (hence, in particular, we have in mind noncompact manifolds). Here, p ∈ ] 1 , ∞ [ p\in ]1,\infty {[} and ℓ = 1 , 2 \ell =1,2 depending on the order of the differential form we are considering. The proof relies on the structure of the Rumin’s complex of differential forms in contact manifolds, on a Sobolev-Gaffney inequality proved by Baldi-Franchi in the setting of the Heisenberg groups and on some geometric properties that can be proved for sub-Riemannian contact manifolds with bounded geometry.

Bounds for eigenfunctions of the Neumann p-Laplacian on noncompact Riemannian manifolds

Abstract Eigenvalue problems for the p-Laplace operator in domains with finite volume, on noncompact Riemannian manifolds, are considered. If the domain does not coincide with the whole manifold, Neumann boundary conditions are imposed. Sharp assumptions ensuring L q {L^{q}} - or L ∞ {L^{\infty}} -bounds for eigenfunctions are offered either in terms of the isoperimetric function or of the isocapacitary function of the domain.

Positive scalar curvature on foliations: The noncompact case

Let (M,gTM) be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and F an integrable subbundle of TM. Let kF be the leafwise scalar curvature associated to gF=gTM|F. We show that if either TM or F is spin, then inf(kF)≤0. This generalizes the famous result of Gromov-Lawson on enlargeable manifolds to the case of foliations. It also extends an ansatz of Gromov on hyper-Euclidean spaces to general enlargeable Riemannian manifolds, as well as recent results on compact enlargeable foliated manifolds due to Benameur-Heitsch et al. to the noncompact situation.

Short-Time Existence for Harmonic Map Heat Flow with Time-Dependent Metrics

In this work, we obtain a short time existence result for harmonic map heat flow coupled with a smooth family of complete metrics in the domain manifold. Our results generalize short time existence results for harmonic map heat flow by Li-Tam [The heat equation and harmonic maps of complete manifolds, Invent. Math., 1991] and Chen-Zhu [Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geometry, 2006]. In particular, we prove the short time existence of harmonic map heat flow along a complete Ricci flow $g(t)$ on $M$ into a complete manifold with curvature bounded from above with a smooth initial map of uniformly bounded energy density, under the assumptions that $|\text{Rm}(g(t))|\leq a/t$ and $g(t)$ is uniformly equivalent to $g(0)$.

Uniqueness of shrinking gradient Kähler–Ricci solitons on non‐compact toric manifolds

We show that, up to biholomorphism, there is at most one complete T n $T^n$ -invariant shrinking gradient Kähler–Ricci soliton on a non-compact toric manifold M. We also establish uniqueness without assuming T n $T^n$ -invariance if the Ricci curvature is bounded and if the soliton vector field lies in the Lie algebra t $\mathfrak {t}$ of T n $T^n$ . As an application, we show that, up to isometry, the unique complete shrinking gradient Kähler–Ricci soliton with bounded scalar curvature on C P 1 × C $\mathbb {C}\mathbb {P}^{1} \times \mathbb {C}$ is the standard product metric associated to the Fubini–Study metric on C P 1 $\mathbb {C}\mathbb {P}^{1}$ and the Euclidean metric on C $\mathbb {C}$ .

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.

Feynman–Kac formula for perturbations of order le 1, and noncommutative geometry

Let Q be a differential operator of order le 1 on a complex metric vector bundle mathscr {E}rightarrow mathscr {M} with metric connection nabla over a possibly noncompact Riemannian manifold mathscr {M}. Under very mild regularity assumptions on Q that guarantee that nabla ^{dagger }nabla /2+Q canonically induces a holomorphic semigroup mathrm {e}^{-zH^{nabla }_{Q}} in Gamma _{L^2}(mathscr {M},mathscr {E}) (where z runs through a complex sector which contains [0,infty )), we prove an explicit Feynman–Kac type formula for mathrm {e}^{-tH^{nabla }_{Q}}, t>0, generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact mathscr {M}’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form TrV~∫0te-sHV∇Pe-(t-s)HV∇ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathrm {Tr}\\left( \\widetilde{V}\\int ^t_0\\mathrm {e}^{-sH^{\ abla }_{V}}P\\mathrm {e}^{-(t-s)H^{\ abla }_{V}}\\mathrm {d}s\\right) , \\end{aligned}$$\\end{document}where V,widetilde{V} are of zeroth order and P is of order le 1. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.

Geometry of positive scalar curvature on complete manifold

Abstract In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. On three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. On higher-dimensional manifold, we obtain a volume growth with a stronger condition.

Secondary representation stability and the once-punctured torus

We study stability patterns in the homology groups of the ordered configuration space of the once-punctured torus. Church and Church–Ellenberg–Farb proved the homology groups of the ordered configuration space of a connected noncompact orientable manifold stabilize in a representation theoretic sense as the number of points in the configuration grows. Miller and Wilson proved that there is a secondary representation stability pattern among the unstable homology classes when viewed as FIM+-modules. We prove that the sequence of minimal generators in the nth-homology of the ordered configuration space of 2n−2 points on the once-punctured torus is neither “free” nor “stably-zero” as an FIM+-module, and that this sequence is generated by the homology classes on at most 4 points. This is the first calculation that shows that secondary representation stability is a non-trivial phenomenon in positive-genus surfaces.

Wasserstein Convergence Rates for Empirical Measures of Subordinated Processes on Noncompact Manifolds

The asymptotic behavior of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than (arXiv:2107.11568), generalize the recent ones in Wang (Stoch Process Appl 144:271–287, 2022) and are shown to be sharp by a typical example. The proof is motivated by the aforementioned works.

Volume growth on manifolds with quadratic curvature decay and its applications

In this paper, we establish global volume growth on an n−dimensional complete noncompact Riemannian manifold whose Ricci curvature is bounded from below byRic(x)≥−(n−1)K01+r2(x), where K0 is a positive constant and r(x)=dist(O,x) is the distance function determined by a fixed point O∈M. Based on which, we derive the Lp−almost boundedness of the Riesz transformation for p∈[1,2] on these manifolds.