Compact Manifold Research Articles

Preserving Positive Intermediate Curvature

Consider a compact manifold N (with or without boundary) of dimension n. Positive m-intermediate curvature interpolates between positive Ricci curvature (m = 1) and positive scalar curvature (m = n-1), and it is obstructed on partial tori N^n = M^{n-m} times mathbb {T}^m. Given Riemannian metrics g, {bar{g}} on (N, partial N) with positive m-intermediate curvature and m-positive difference h_g - h_{{bar{g}}} of second fundamental forms we show that there exists a smooth family of Riemannian metrics with positive m-intermediate curvature interpolating between g and {bar{g}}. Moreover, we apply this result to prove a non-existence result for partial torical bands with positive m-intermediate curvature and strictly m-convex boundaries.

On Schrödinger operators modified by [formula omitted] interactions

We study the spectral properties of a Schrödinger operator H0 modified by δ interactions and show explicitly how the poles of the new Green’s function are rearranged relative to the poles of original Green’s function of H0. We prove that the new bound state energies are interlaced between the old ones, and the ground state energy is always lowered if the δ interaction is attractive. We also derive an alternative perturbative method of finding the bound state energies and wave functions under the assumption of a small coupling constant in a somewhat heuristic manner. We further show that these results can be extended to cases in which a renormalization process is required. We consider the possible extensions of our results to the multi center case, to δ interaction supported on curves, and to the case, where the particle is moving in a compact two-dimensional manifold under the influence of δ interaction. Finally, the semi-relativistic extension of the last problem has been studied explicitly.

Geometry of static perfect fluid space-time

In this article, we investigate the geometry of static perfect fluid space-time (SPFST) on compact manifolds with boundary. We use the generalized Reilly’s formula to establish a geometric inequality for a SPFST involving the area of the boundary and its volume. Moreover, we obtain new boundary estimates for SPFST. One of the boundary estimates is obtained in terms of the Brown–York mass and another one related to the first eigenvalue of the Jacobi operator. In addition, we provide a new (simply connected) counterexample to the Cosmic no-hair conjecture for arbitrary dimension

Nodal Sets of Eigenfunctions of Sub-Laplacians

Abstract Nodal sets of eigenfunctions of elliptic operators on compact manifolds have been studied extensively over the past decades. In this note, we initiate the study of nodal sets of eigenfunctions of hypoelliptic operators on compact manifolds, focusing on sub-Laplacians. A standard example is the sum of squares of bracket-generating vector fields on compact quotients of the Heisenberg group. Our results show that nodal sets behave in an anisotropic way, which can be analyzed with standard tools from sub-Riemannian geometry such as sub-Riemannian dilations, nilpotent approximation, and desingularization at singular points. Furthermore, we provide a simple example demonstrating that for sub-Laplacians, the Hausdorff measure of nodal sets of eigenfunctions cannot be bounded above by $\sqrt \lambda $, which is the bound conjectured by Yau for Laplace–Beltrami operators on smooth manifolds.

Strichartz estimates for the half wave/Klein–Gordon and Dirac equations on compact manifolds without boundary

In this paper we study Strichartz estimates for the half wave, the half Klein–Gordon and the Dirac Equations on compact manifolds without boundary, proving in particular for each of these flows local in time estimates both for the wave and Schrödinger admissible couples (in this latter case with an additional loss of regularity). The strategy for the proof is based on a refined version of the WKB approximation.

On Bott–Chern and Aeppli cohomologies of almost complex manifolds and related spaces of harmonic forms

In this paper we introduce several new cohomologies of almost complex manifolds, among which stands a generalization of Bott–Chern and Aeppli cohomologies defined using the operators d, dc. We explain how they are connected to already existing cohomologies of almost complex manifolds and we study the spaces of harmonic forms associated to d, dc, showing their relation with Bott–Chern and Aeppli cohomologies and to other well-studied spaces of harmonic forms. Notably, Bott–Chern cohomology of 1-forms is finite-dimensional on compact manifolds and provides an almost complex invariant hd+dc1 that distinguishes between almost complex structures. On almost Kähler 4-manifolds, the spaces of harmonic forms we consider are particularly well-behaved and are linked to harmonic forms considered by Tseng and Yau in the study of symplectic cohomology.

Torsion in cohomology and dimensional reduction

Conventional wisdom dictates that ℤN factors in the integral cohomology group Hp(Xn, ℤ) of a compact manifold Xn cannot be computed via smooth p-forms. We revisit this lore in light of the dimensional reduction of string theory on Xn, endowed with a G-structure metric that leads to a supersymmetric EFT. If massive p-form eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to them will have a non-trivial smeared delta form, that is an EFT long-wavelength description of p-form currents of the (n − p)-cycles of Xn. We conjecture that, whenever torsion cycles are calibrated, their linking number can be computed via their smeared delta forms. From the EFT viewpoint, a torsion factor in cohomology corresponds to a ℤN gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.

Positive scalar curvature on manifolds with fibered singularities

Abstract A (compact) manifold with fibered 𝑃-singularities is a (possibly) singular pseudomanifold M Σ M_{\Sigma} with two strata: an open nonsingular stratum M ̊ \mathring{M} (a smooth open manifold) and a closed stratum β ⁢ M \beta M (a closed manifold of positive codimension), such that a tubular neighborhood of β ⁢ M \beta M is a fiber bundle with fibers each looking like the cone on a fixed closed manifold 𝑃. We discuss what it means for such an M Σ M_{\Sigma} with fibered 𝑃-singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type 𝑃 is either Z / k \mathbb{Z}/k or S 1 S^{1} , and 𝑀 and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin𝑐 manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin𝑐 line bundle.

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.