Manifold Of Nonpositive Sectional Curvature Research Articles

The coarse geometric Novikov conjecture for subspaces of non-positively curved manifolds

In this paper, we prove the coarse geometric Novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete Riemannian manifold of non-positive sectional curvature.

An Estimate for the Entropy of Hamiltonian Flows

In this paper, we present a generalization to Hamiltonian flows on symplectic manifolds of the estimate proved by Ballmann and Wojtkovski in [4] for the dynamical entropy of the geodesic flow on a compact Riemannian manifold of nonpositive sectional curvature. Given such a Riemannian manifold M, Ballmann and Wojtkovski proved that the dynamical entropy h ? of the geodesic flow on M satisfies the inequality $h_{\mu } \geqslant {\int\limits_{SM} {{\text{Tr}}{\sqrt { - K{\left( v \right)}} }d\mu {\left( v \right)}} },$ where v is a unit vector in T p M if p is a point in M, SM is the unit tangent bundle on M, K(v) is defined as $K{\left( \upsilon \right)} = {\user1{\mathcal{R}}}{\left( { \cdot ,\upsilon } \right)}\upsilon $ , where ${\user1{\mathcal{R}}}$ is the Riemannian curvature of M, and ? is the normalized Liouville measure on SM. We consider a symplectic manifold M of dimension 2n, and a compact submanifold N of M, given by the regular level set of a Hamiltonian function on M; moreover, we consider a smooth Lagrangian distribution on N, and we assume that the reduced curvature $\hat{R}_z^h$ of the Hamiltonian vector field $\vec{h}$ with respect to the distribution is non-positive. Then we prove that under these assumptions, the dynamical entropy h ? of the Hamiltonian flow with respect to the normalized Liouville measure on N satisfies $h_{\mu } \geqslant {\int\limits_N {{\text{Tr}}{\sqrt { - \ifmmode\expandafter\hat\else\expandafter\^\fi{R}^{h}_{z} } }d\mu } }.$

The double cover relative to a convex domain and the relative isoperimetric inequality

AbstractWe prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

Perturbations of the Harmonic Map Equation

We consider perturbations of the harmonic map equation in the case where the target manifold is a closed Riemannian manifold of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.

Behavior of the Double-Layer Potential for a Parabolic Equation on a Manifold

We prove that, similarly to the double-layer potential in $$\mathbb{R}^n $$ , the double-layer potential constructed in a Riemann manifold of nonpositive sectional curvature has a jump in passing through the surface where its density is defined.

Proximal Point Algorithm On Riemannian Manifolds

In this paper we consider the minimization problem with constraints. We will show that if the set of constraints is a Riemannian manifold of nonpositive sectional curvature, and the objective function is convex in this manifold, then the proximal point method in Euclidean space is naturally extended to solve that class of problems. We will prove that the sequence generated by our method is well defined and converge to a minimizer point. In particular we show how tools of Riemannian geometry, more specifically the convex analysis in Riemannian manifolds, can be used to solve nonconvex constrained problem in Euclidean, space.

Perturbations of the harmonic map equation

We consider perturbations of the harmonic map equation in the case where the target manifold is a closed Riemannian manifold of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincar e in

Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature

Let \(M^{2n} \) be a compact Riemannian manifold of non-positive sectional curvature. It is shown that if \(M^{2n}\) is homeomorphic to a Kahler manifold, then its Euler number satisfies the inequality \( (-1)^n \chi(M^{2n})\geq 0\).

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kahler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kahler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.

Limiting angles of Γ-martingales

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m≥ 3 and that, outside a fixed compact set, the sectional curvatures are bounded above by −c 1/{r 2 ln r} and below by −c 2 r 2, where c 1 and c 2 are two positive constants and r is the geodesic distance from a fixed point. We show that, when κ≥ 1 satisfies certain conditions, the angular part of a κ-quasi-conformal Γ-martingale on M tends to a limit as time tends to infinity and the closure of the support of the distribution of this limit is the entire sphere at infinity. This improves both a result of Le for Brownian motion and also results concerning the non-existence of κ-quasi-conformal harmonic maps from certain types of Riemannian manifolds into M.

Topological entropy of semi-dispersing billiards

In this paper we continue to explore the applications of the connections between singular Riemannian geometry and billiard systems that were first used in [6] to prove estimates on the number of collisions in non-degenerate semi-dispersing billiards.In this paper we show that the topological entropy of a compact non-degenerate semi-dispersing billiard on any manifold of non-positive sectional curvature is finite. Also, we prove exponential estimates on the number of periodic points (for the first return map to the boundary of a simple-connected billiard table) and the number of periodic trajectories (for the billiard flow). In \S5 we prove some estimates for the topological entropy of Lorentz gas.

Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature

Let ( M,g) be a smooth compact Riemannian n-manifold, n ≥ 2, let p(1, n) real, and let H 1 p ( M) be the standard Sobolev space of order p. By the Sobolev embedding theorem, H 1 p ( M) ⊂ L p * ( M) where p * = np/( n - p). Classically, this leads to some Sobolev inequality (I p 1), and then to some Sobolev inequality (I p p ) where each term in (I p 1) is elevated to the power p. Long standing questions were to know if the optimal versions with respect to the first constant of (I p 1) and (I p p ) do hold. Such questions received an affirmative answer by Hebey-Vaugon for p = 2, and on what concerns (I p 1), by Aubin for two-dimensional manifolds and for manifolds of constant sectional curvature. Recently, Druet proved that for p > 2, and p 2 < n, the optimal version of (I p p ) is false if the scalar curvature of g is positive somewhere, while for p > 1, the optimal version of (I p p ) does hold on flat torii and compact hyperbolic spaces. We prove here that the optimal version of (I p p ), p > 1, does hold for compact manifolds of nonpositive sectional curvature in any dimension where the Cartan-Hadamard conjecture is true. In particular, since the Cartan-Hadmard conjecture is true in dimensions 2, 3, and 4, the optimal version of (I p p ) does hold on any compact manifold of nonpositive sectional curvature of dimension 2, 3, or 4.

Rough isometry and the asymptotic Dirichlet problem

We propose a new asymptotic Dirichlet problem for harmonic functions via the rough isometry on a certain class of Riemannian manifolds. We prove that this problem is solvable for naturally defined class of functions. This result generalizes those of Schoen and Yau and of Cheng. 1. Introduction. The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. It has a long history, and by now, it is well understood by the works of the first author, M. Anderson, D. Sullivan, R. Schoen and others, when M is a Cartan-Hadamar d manifold with sectional curvature — b2<KM< —a2<0. (By a Cartan-Hadamar d manifold, we mean a complete simply connected manifold of nonpositive sectional curvature.) In (Ch), the first author posed the asymptotic Dirichlet problem and proved that it is solvable when a Cartan-Hadamar d manifold M with sectional curvature KM<—a2<0 satisfies the convex conic neighborhood condition. In (A), Anderson constructed a convex neighborhood, thereby solving the problem when the sectional curvature satisfies — b2<KM< — a2<0. At the same time, Sullivan (S) also solved this problem using the probabilistic approach. In (A-S), Anderson and Schoen showed that the Martin boundary of M can be identified with M(oo) which is naturally defined to be the set of the asymptotic classes of geodesic rays. By constructing the Poisson kernel, they obtained the representation formula for harmonic functions, and proved the Fatou-type theorem. The essence of all of the above works is that the curvature assumption enables one to control the angle via the Toponogov comparison theorem and the convexity property near the boundary at infinity M(oo). There have been many attempts to generalize the above results. A typical approach is to relax the curvature assumption to allow curvature decay at a certain rate. But the basic method of proof still remains the same and the improvements are mostly techni- cal. One interesting generalization that does not directly involve the curvature bound

Rigidity of Einstein manifolds of nonpositive curvature

We give sufficient conditions for a compact Einstein manifold of nonpositive sectional curvature to be isometric to a hyperbolic manifold if they have isomorphic fundamental groups. We prove a finiteness result for Einstein metrics on a compact manifold of negative sectional curvature. Spectral rigidity of locally symmetric spaces of negative curvature is considered.

Limiting angle of Brownian motion on certain manifolds

Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m ≧ 3. If, outside a fixed compact set, the sectional curvatures are bounded above by a negative constant multiple of the inverse of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of the geodesic distance, then the angular part of Brownian motion on M tends to a limit as time tends to infinity, and the closure of the support of the distribution of this limit is the entire Sm−1. This improves a result of Hsu and March.

Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification of finite volume quotients of the complex hyperbolic space.

A boundary value problem for Hermitian harmonic maps and applications

We study the existence and uniqueness problems for Hermitian harmonic maps from Hermitian manifolds with boundary to Riemannian manifolds of nonpositive sectional curvature and with convex boundary. The complex analyticity of such maps and the related rigidity problems are also investigated.

Symplectomorphic codimension 1 totally geodesic submanifolds

We show that the coisotropic totally geodesic properly embedded submanifolds of codimension 1 of a simply connected complete Kähler manifold of non-positive sectional curvature are symplectically linearizable.

Rank rigidity for foliations by manifolds of nonpositive curvature

We develop a rank rigidity theorem for finite volume foliations by manifolds of nonpositive sectional curvature. It implies that if the leaves are irreducible Hadamard manifolds of geometric rank ⩾ 2, then the leaves are symmetric.

Horofunctions, busemann functions, and ideal boundaries of open manifolds of nonnegative curvature

1. Definitions An open manifold X n is defined to be a complete noncompact manifold. The metric on X ~ will be denoted by d. Open manifolds are obtainable, for example, as follows: Let ~n be a compact submanifold in R N with boundary and let f be a continuous function on ~n whose zero set coincides with the boundary 0X '~. Construct an embedding r of the interior of ~n into the space R x x R by assigning r = (x,f.-l(x)), x 6 Int(X'~). In this case the boundary 0)( '~ "goes to infinity" and the set X n = ~(Int(X'~)) endowed with the metric induced by the embedding is an open manifold diffeomorphic to the interior of the compact manifold ~-,z with boundary. For an open manifold X n there are known several ways of constructing the ideal boundary that corresponds to 0)~ r~ in the example presented above (see the definitions of the sets O(X), B(X), and X(oc) below). If X is an "Hadamard manifold", i.e. a simply connected manifold of nonpositive sectional curvature, then all the definitions of ideal boundary determine pairwise homeomorphic sets It]. In the present article we restrict ourselves to the case of manifolds of nonnegative or almost nonnegative curvature. We will exhibit an example of a surface X for which the above-indicated definitions give pairwise nonhomeomorphic sets, namely: the space of horofunctions O(X) is homeomorphic to a closed interval, the space B(X) of Busemann functions is a two-point set, and the space X(oo) of equivalence classes of rays is a singleton. On the exhibited surface there are exactly two poles. The same two points are the unique souls of X (see the definition of a soul of an open manifold of nonnegative curvature in_ [2]), and the points of the shortest geodesic joining them are the minimum points of the horofunctions. The correspondence indicated is one-to-one. The surface constructed is a space of nonnegative curvature in the sense of the comparison theorem but is not a smooth Cl-surface. However, it admits of approximation in the Hausdorff metric by smooth surfaces and since the convergence of surfaces in the Hausdorff metric implies the convergence of their ideal boundaries, the example constructed ensures existence of smooth surfaces with pairwise nonhomeomorphic ideal boundaries. 1.1. Rays and Busemann functions. Any open manifold is unbounded. Therefore, for every point p, there exists a sequence of points pi such that d(p, Pl) -+ oo as i --+ ~c. Connect the points p and pi by the shortest geodesics li(s) as follows: From the sequence {i} one can choose a subsequence {j} such that the sequence constituted by the directions of the shortest geodesics lj(s) at the point p = lj(O) converges to some vector v. Then every segment of the geodesic l(s) = expp(sv) is a limit of the corresponding segments of the geodesics lj(s). By that reason, the geodesic l(s), 0 <_" s < co, is a shortest geodesic on each of its segments. Such a geodesic is calIed a ray. The set of all rays issuing from the point p will be denoted by 77.p. The geodesic l(s), -oo < s < cc (infinite to both sides), is called a line in case it is a shortest geodesic over each of its segments. DEFINITION 1.1. Let l(s), 0 <_ s < co, be a ray. The function