Compact Manifold Research Articles

Cartan actions of higher rank abelian groups and their classification

We study R k × Z ℓ \mathbb {R}^k \times \mathbb {Z}^\ell actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program.

Finite Groups Acting on Compact Complex Parallelizable Manifolds

Abstract We prove that the automorphism group of a compact complex parallelizable manifold is Jordan. In the course of the proof, we show that outer automorphism groups of cocompact lattices in complex Lie groups have bounded finite subgroups.

Conformal perturbations of modified Novikov operators and the Kastler–Kalau–Walze type theorem

In this paper, we establish two Kastler–Kalau–Walze type theorems for conformal perturbations of modified Novikov operators on four-dimensional and six-dimensional compact manifolds with (respectively, without) boundary.

A tensor product approach to non-local differential complexes

We study differential complexes of Kolmogorov–Alexander–Spanier type on metric measure spaces associated with unbounded non-local operators, such as operators of fractional Laplacian type. We define Hilbert complexes, observe invariance properties and obtain self-adjoint non-local analogues of Hodge Laplacians. For d-regular measures and operators of fractional Laplacian type we provide results on removable sets in terms of Hausdorff measures. We prove a Mayer–Vietoris principle and a Poincaré lemma and verify that in the compact Riemannian manifold case the deRham cohomology can be recovered.

On recurrent Riemannian and Ricci curvatures of Finsler metrics

In this paper, we study a property of Riemannian and Ricci curvatures under which it reproduces itself, namely, recurrent Finsler metrics. We prove that if (M,F) is a recurrent Finsler manifold of non-zero isotropic flag curvature, then F is a Landsberg metric. It follows that Every positively complete 2-dimensional Randers metric is recurrent if and only if it is a Riemannian or locally Minkowskian metric. Next, we study two positive (or negative) projectively related Ricci parallel Finsler metrics on a compact manifold. We show that the projective equivalence is trivial and then the Riemannian curvatures are equal. In the same vein, we explore the class of Ricci-recurrent Randers metrics with closed and conformal form, and show that the related Riemannian metric is Ricci-recurrent if and only if the Randers metric is a Berwald metric. Finally, we find the necessary and sufficient condition under which a Kropina metric be Ricci-recurrent, provided that its one-form is closed and conformal.

Complex symplectic Lie algebras with large Abelian subalgebras

We present two constructions of complex symplectic structures on Lie algebras with large Abelian ideals. In particular, we completely classify complex symplectic structures on almost Abelian Lie algebras. By considering compact quotients of their corresponding connected, simply connected Lie groups we obtain many examples of complex symplectic manifolds which do not carry (hyper)kähler metrics. We also produce examples of compact complex symplectic manifolds endowed with a fibration whose fibers are Lagrangian tori.

The total Cartan curvature of Reinhardt surfaces

The Cartan umbilical tensor is a crucial biholomorphic invariant for three-dimensional strictly pseudoconvex CR manifolds. When considering compact manifolds, its L1-norm, referred to as the total Cartan curvature, is an important global numerical invariant. In this study, we seek to calculate the total Cartan curvature of T3, the three-torus, which features a specific T2-invariant CR structure, incorporating Reinhardt boundaries with closed logarithmic generating curves as a special instance. Our findings present an unexpected corollary indicating that, for many of these boundaries, the total Cartan curvature is equal to 8π2 times the Burns–Epstein invariant.

Functionals for the study of lcK metrics on compact complex manifolds

We propose an approach to the existence problem for locally conformally Kähler metrics on compact complex manifolds by introducing and studying a functional that is different according to whether the complex dimension of the manifold is 2 or higher.

Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on N-dimensional RCD(K,N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{RCD}(K,N)$$\\end{document} spaces (X,d,HN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(X,\ extsf {d},\\mathscr {H}^N)$$\\end{document}. Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements seem to be new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.

Computing a Link Diagram From Its Exterior

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, and allows us to recover links with hundreds of crossings. We use it to find the first diagrams known for 23 principal congruence arithmetic link exteriors; the largest has over 2500 crossings. Other applications include finding pairs of knots with the same 0-surgery, which relates to questions about slice knots and the smooth 4D Poincaré conjecture.

The topology of compact rank-one ECS manifolds

AbstractPseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank of a certain distinguished null parallel distribution $\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a twofold isometric covering, must be a bundle over the circle with leaves of $\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally homogeneous case if one assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold, the leaves of $\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.

Kirchhoff equations with strong singularity in closed manifolds

In this paper, we study a class of Kirchhoff-type problems involving strong singular nonlinearities on compact Riemannian manifolds. With the help of variational method and Nehari method, we prove that this problem has a unique positive solution.

Determining Lamé coefficients by the elastic Dirichlet-to-Neumann map on a Riemannian manifold

For the Lamé operator with variable coefficients λ and µ on a smooth compact Riemannian manifold (M, g) with smooth boundary , we give an explicit expression for the full symbol of the elastic Dirichlet-to-Neumann map . We show that uniquely determines the partial derivatives of all orders of the Lamé coefficients λ and µ on . Moreover, for a nonempty smooth open subset , suppose that the manifold and the Lamé coefficients are real analytic up to Γ, we prove that uniquely determines the Lamé coefficients on the whole manifold .

DLR–KMS correspondence on lattice spin systems

The Dobrushin–Lanford–Ruelle condition (Dobrushin in Theory Prob Appl 17:582–600, 1970. https://doi.org/10.1137/1115049; Lanford and Ruelle in Commun Math Phys 13:194–215, 1969. https://doi.org/10.1007/BF01645487) and the classical Kubo–Martin–Schwinger (KMS) condition (Gallavotti and Verboven in Nuov Cim B 28:274–286, 1975. https://doi.org/10.1007/BF02722820) are considered in the context of classical lattice systems. In particular, we prove that these conditions are equivalent for the case of a lattice spin system with values in a compact symplectic manifold by showing that infinite-volume Gibbs states are in bijection with KMS states.

Quantitative estimates for fractional Sobolev mappings in rational homotopy groups

Let N⊂RM be a smooth simply connected compact manifold without boundary. A rational homotopy subgroup of πN(N) is represented by a homomorphism deg:πN(N)→R.For maps f:SN→N we give a quantitative estimate of its rational homotopy group element deg([f])∈R in terms of its fractional Sobolev-norm. That is, we show that for all β∈(β0(deg),1], |deg([f])|≤C(deg)[f]Wβ,Nβ(SN)N+L(deg)β.Here C(deg)>0, L(deg)∈N, β0(deg)∈(0,1) are computable from the rational homotopy group represented by deg. This extends earlier work by Van Schaftingen and the second author on the Hopf degree to the Novikov’s integral representation for rational homotopy groups as developed by Sullivan, Novikov, Hardt and Rivière.

Quantum field theory of physical and purely virtual particles in a finite interval of time on a compact space manifold: diagrams, amplitudes and unitarity

We provide a diagrammatic formulation of perturbative quantum field theory in a finite interval of time τ, on a compact space manifold Ω. We explain how to compute the evolution operator U(tf, ti) between the initial time ti and the final time tf = ti + τ, study unitarity and renormalizability, and show how to include purely virtual particles, by rendering some physical particles (and all the ghosts, if present) purely virtual. The details about the restriction to finite τ and compact Ω are moved away from the internal sectors of the diagrams (apart from the discretization of the three-momenta), and coded into external sources. Unitarity is studied by means of the spectral optical identities, and the diagrammatic version of the identity U†(tf, ti)U(tf, ti) = 1. The dimensional regularization is extended to finite τ and compact Ω, and used to prove, under general assumptions, that renormalizability holds whenever it holds at τ = ∞, Ω = ℝ3. Purely virtual particles are introduced by removing the on-shell contributions of some physical particles, and the ghosts, from the core diagrams, and trivializing their initial and final conditions. The resulting evolution operator Uph(tf, ti) is unitary, but does not satisfy the more general identity Uph(t3, t2)Uph(t2, t1) = Uph(t3, t1). As a consequence, Uph(tf, ti) cannot be derived from a Hamiltonian in a standard way, in the presence of purely virtual particles.

A Note on Incompressible Vector Fields

In this paper, we use incompressible vector fields for characterizing Killing vector fields. We show that on a compact Riemannian manifold, a nontrivial incompressible vector field has a certain lower bound on the integral of the Ricci curvature in the direction of the incompressible vector field if, and only if, the vector field ξ is Killing. We also show that a nontrivial incompressible vector field ξ on a compact Riemannian manifold is a Jacobi-type vector field if, and only if, ξ is Killing. Finally, we show that a nontrivial incompressible vector field ξ on a connected Riemannian manifold has a certain lower bound on the Ricci curvature in the direction of ξ, and if ξ is also a geodesic vector field, it necessarily implies that ξ is Killing.

Rigidity results for the p‐Laplace type equations on compact Riemannian manifolds

In this paper, we obtain two rigidity results for ‐Laplace type equation and ‐Laplace equation with exponential nonlinearity on ‐dimensional compact Riemannian manifolds by using of nonlinear flow and the carré du champ methods, respectively, where rigidity means that the PDE has only constant solution when a parameter is in a certain range. Moreover, an interpolation inequality is derived as an application.

Poincaré metric of holomorphic foliations with non-degenerate singularities

Consider a Brody hyperbolic foliation [Formula: see text] with non-degenerate singularities on a compact complex manifold. We show that its leafwise Poincaré metric is transversally Hölder continuous with a logarithmic slope towards the singular set of [Formula: see text].

A-foliations of codimension two on compact simply-connected manifolds

We show that a singular Riemannian foliation of codimension two on a compact simply-connected Riemannian (n+2)-manifold, with regular leaves homeomorphic to the n-torus, is given by a smooth effective n-torus action. This solves in the negative for the codimension 2 case a question about the existence of foliations by exotic tori on simply-connected manifolds.