Nonpositive Sectional Curvature Research Articles

Tensor Tomography on Negatively Curved Manifolds of Low Regularity

We prove solenoidal injectivity for the geodesic X-ray transform of tensor fields on simple Riemannian manifolds with C1,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,1}$$\\end{document} metrics and non-positive sectional curvature. The proof of the result rests on Pestov energy estimates for a transport equation on the non-smooth unit sphere bundle of the manifold. Our low regularity setting requires keeping track of regularity and making use of many functions on the sphere bundle having more vertical than horizontal regularity. Some of the methods, such as boundary determination up to gauge and regularity estimates for the integral function, have to be changed substantially from the smooth proof. The natural differential operators such as covariant derivatives are not smooth.

Rellich inequalities via Riccati pairs on model space forms

We present a simple method for proving Rellich inequalities on Riemannian manifolds with constant, non-positive sectional curvature. The method is built upon simple convexity arguments, integration by parts, and the so-called Riccati pairs, which are based on the solvability of a Riccati-type ordinary differential inequality. These results can be viewed as the higher order counterparts of the recent work by Kajántó, Kristály, Peter, and Zhao, discussing Hardy inequalities using Riccati pairs.

The twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1→N→G→Q→1, we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G.

Eigenfunction restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature

For 2 ≤ p > 4 2\leq p>4 , we study the L p L^p norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact 2 2 -dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov [Duke Math. J. 138 (2007), pp. 445–486], and Hu [Forum Math. 21 (2009), pp. 1021–1052] found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For p = 4 p=4 , we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [Comm. Math. Phys. 350 (2017), pp. 1299–1325], Sogge [Math. Res. Lett. 24 (2017), pp. 549–570], and Bourgain [Geom. Funct. Anal. 1 (1991), pp. 147–187].

Plurisubharmonicity of the Dirichlet energy and deformations of polarized manifolds

We show that if [Formula: see text] is a polarized family of compact Kähler manifolds over the open unit disk [Formula: see text], if [Formula: see text] is a Riemannian manifold of nonpositive complexified sectional curvature, and if [Formula: see text] is a smooth family of pluriharmonic maps, then the Dirichlet energy [Formula: see text] is a subharmonic function of [Formula: see text].

Liouville theorems for ancient solutions to the [formula omitted]-harmonic map heat flows

We investigate the Liouville properties of ancient solutions to the V-harmonic map heat flows from complete noncompact Riemannian manifolds with nonnegative Bakry–Emery Ricci curvature. When the target is a simply connected complete Riemannian manifold with nonpositive sectional curvature, then a Liouville theorem of ancient solutions holds under certain growth condition near infinity. When the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, we show that if the image of the ancient solution u is contained in a regular ball near infinity, then u is a constant.

ON SOME MANIFOLDS OF CONSTANT NEGATIVE CURVATURE

Manifold of constant negative curvature a great role in the field of physics, mathematics and engineering because it paves to the knowledge Gaussian curvature , n-sphere(s) is a topological n-manifold and objects of constant negative curvature are less familiar but they do appear in nature in the shape of corals and leave . No surprisingly , its plays an important role in geometric topology. The study aims to explain a generalization manifold of constant negative curvature . WE followed the analytical induction mathematical method . We found the following some results. Manifold of constant negative curvature indicates to know the behavior of some of the functions and also it reveals the Cartan-Hadamard theorem which is considered one of the importance aims of simply-connected manifold of nonpositive sectional curvature.

Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques

We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings’ foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2 2 -dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are as short as possible (length two words), finitely generated free subgroups, non-finitely presentable subgroups, and subgroups of fundamental groups of square complexes of nonpositive sectional curvature.

On Almost Nonpositive k-Ricci Curvature

Motivated by the recent work of Chu–Lee–Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive k-Ricci curvature, we consider a natural notion of almost nonpositive k-Ricci curvature, which is weaker than the existence of a Kähler metric with nonpositive k-Ricci curvature. When \(k=1\), this is just the almost nonpositive holomorphic sectional curvature introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with k-Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact Kähler manifold of almost nonpositive k-Ricci curvature must have nef canonical line bundle.

A Contribution of Liouville-Type Theorems to the Geometry in the Large of Hadamard Manifolds

A complete, simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. In this article, we prove Liouville-type theorems for isometric and harmonic self-diffeomorphisms of Hadamard manifolds, as well as Liouville-type theorems for Killing–Yano, symmetric Killing and harmonic tensors on Hadamard manifolds.

On Global Dynamics of Schrödinger Map Flows on Hyperbolic Planes Near Harmonic Maps

The results of this paper are twofold. In the first part, we prove that for Schrödinger map flows from hyperbolic planes to Riemannian surfaces with non-positive sectional curvatures, the harmonic maps which are holomorphic or anti-holomorphic of arbitrary size are asymptotically stable. In the second part, we prove that for Schrödinger map flows from hyperbolic planes into Kähler manifolds, the admissible harmonic maps of small size are asymptotically stable. The asymptotic stability results stated here contain two types: one is the convergence in $$L^{\infty }_x$$ as the previous works, the other is convergence to harmonic maps plus radiation terms in the energy space, which is new in literature of Schrödinger map flows without symmetric assumptions.

The Wu–Yau theorem on Sasakian manifolds

In this paper, we proved that a compact Sasakian manifold [Formula: see text] with negative transverse holomorphic sectional curvature must have a Sasakian structure [Formula: see text] with negative transverse Ricci curvature. Similarly, a compact Sasakian manifold with nonpositive transverse holomorphic sectional curvature, then the negative first basic Chern class [Formula: see text] is transverse nef and we have the Miyaoka–Yau-type inequality. When transverse holomorphic sectional curvature is quasi-negative, we obtain a Chern number inequality.

CARLSON–GRIFFITHS THEORY FOR COMPLETE KÄHLER MANIFOLDS

AbstractWe investigate Carlson–Griffiths’ equidistribution theory of meormorphic mappings from a complete Kähler manifold into a complex projective algebraic manifold. By using a technique of Brownian motions developed by Atsuji, we obtain a second main theorem in Nevanlinna theory provided that the source manifold is of nonpositive sectional curvature. In particular, a defect relation follows if some growth condition is imposed.

Broken ray tensor tomography with one reflecting obstacle

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and vanishes on the rest. This is done in a geometry with non-positive sectional curvature and a strictly convex obstacle in any dimension. We give two proofs, both of which contain new features also in the absence of reflections. The result is new even for scalars in dimensions above two.

Price inequalities and Betti number growth on manifolds without conjugate points

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points. Finally, we give a vanishing result for $L^{2}$-Betti numbers of closed manifolds without conjugate points.

Higher genera for proper actions of Lie groups, II: The case of manifolds with boundary

Let G be a finitely connected Lie group and let K be a maximal compact subgroup. Let M be a cocompact G-proper manifold with boundary, endowed with a G-invariant metric which is of product type near the boundary. Under additional assumptions on G, for example that it satisfies the Rapid Decay condition and is such that G/K has nonpositive sectional curvature, we define higher Atiyah-Patodi-Singer C^*-indices associated to smooth group cocycles on G and to a generalized G-equivariant Dirac operator D on M with L^2-invertible boundary operator D_\partial. We then establish a higher index formula for these C^*-indices and use it in order to introduce higher genera for M, thus generalizing to manifolds with boundary the results that we have established in Part 1. Our results apply in particular to a semisimple Lie group G. We use crucially the pairing between suitable relative cyclic cohomology groups and relative K-theory groups.

Classification of static asymptotically flat spacetimes with a photon sphere in Einstein-multiple-scalar field theory

In the present paper we consider the Einstein-multiple-scalar field theory. When the target space of the scalar fields is a complete, simply connected Riemannian manifold with non-positive sectional curvature we prove that the static and asymptotically flat solutions which posses a photon sphere are uniquely specified by their asymptotic data, i.e. by their mass, scalar charges and asymptotic values of the scalar fields. The unique solution with a photon sphere and prescribed asymptotic data is explicitly constructed.

Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds with negative curvature

In this paper we obtain Hardy, weighted Trudinger-Moser and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants on Riemannian manifolds with non-positive sectional curvature and, in particular, a variety of new estimates on hyperbolic spaces. Moreover, in some cases we also show their equivalence with Trudinger-Moser inequalities. As consequences, the relations between the constants of these inequalities are investigated yielding asymptotically best constants in the obtained inequalities. We also obtain the corresponding uncertainty type principles.

The space of almost calibrated (1,1)–forms on a compact Kähler manifold

The space $\mathcal{H}$ of "almost calibrated" $(1,1)$ forms on a compact K\"ahler manifold plays an important role in the study of the deformed Hermitian-Yang-Mills equation of mirror symmetry as emphasized by recent work of the second author and Yau, and is related by mirror symmetry to the space of positive Lagrangians studied by Solomon. This paper initiates the study of the geometry of $\mathcal{H}$. We show that $\mathcal{H}$ is an infinite dimensional Riemannian manifold with non-positive sectional curvature. In the hypercritical phase case we show that $\mathcal{H}$ has a well-defined metric structure, and that its completion is a ${\rm CAT}(0)$ geodesic metric space, and hence has an intrinsically defined ideal boundary. Finally, we show that in the hypercritical phase case $\mathcal{H}$ admits $C^{1,1}$ geodesics, improving a result of the second author and Yau. Using results of Darvas-Lempert we show that this result is sharp.

Path and quasi-homotopy for Sobolev maps between manifolds

We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are quasi-homotopic – and observe the rather surprising fact that quasi-homotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with non-positive sectional curvature, and the contrasting case of the target being a sphere.