N-dimensional Compact Riemannian Manifold Research Articles

ON THE COMPACT RIEMANNIAN MANIFOLDS WITH SOME GEODESICAL PROPERTIES

In the paper, we study an n-dimensional compact Riemannian manifold (M, g) with the property that the lengthes of the images in M of any geodesic curves are finite.

Rigidity of amalgamated products in negative curvature

Let $\Gamma$ be the fundamental group of a compact n-dimensional riemannian manifold X of sectional curvature bounded above by -1. We suppose that $\Gamma$ is a free product of its subgroup A and B over the amalgamated subgroup C. We prove that the critical exponent $\delta(C)$ of C satisfies $\delta(C) \geq n-2$. The equality happens if and only if there exist an embedded compact hypersurface Y in X , totally geodesic, of constant sectional curvature -1, with fundamental group C and which separates X in two connected components whose fundamental groups are A and B. Similar results hold if $\Gamma$ is an HNN extension, or more generally if $\Gamma$ acts on a simplicial tree without fixed point.

A Lidskii type formula for Dixmier traces

We present a new formula to compute Dixmier traces τ ω ( T ) of pseudodifferential operators (respectively, almost periodic pseudodifferential operators) of order − n on n-dimensional compact Riemannian manifolds (respectively, R n ). Under a natural condition on the operator T, we show that τ ω ( T ) = ω - lim t → ∞ 1 log ( 1 + t ) ∫ λ ∉ 1 t G λ d μ T ( λ ) , where G is any bounded neighborhood of 0 ∈ C and μ T is the Brown spectral measure of T. If T is measurable, then the ω-limit may be replaced with the true (ordinary) limit. Our approach works equally well in both type I and II settings. To cite this article: N.A. Azamov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe

Let (M, g) be an n-dimensional compact smooth Riemannian manifold (without boundary). For n=2, we know from the uniformization theorem of Poinca% that there exist metrics that are pointwise conformal to g and have constant Gauss curvature. For n~>3, the well-known Yamabe conjecture states that there exist metrics which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture is proved through the work of Yamabe [65], Trudinger [58], Aubin [2] and Schoen [53]. The Yamabe and related problems have attracted much attention in the last 30 years or so, see, e.g., [57], [3] and the references therein. Important methods and techniques in overcoming loss of compactness have been developed in such studies, which also play important roles in the research of other areas of mathematics. For n~>3, let ~=u4/(n-2)g, where u is some positive function on M. The scalar curvature RO of ~ can be calculated as

A lower bound for first eigenvalue with mixed boundary condition

Let M be an n-dimensional compact Riemannian manifold with or without boundary, and its Ricci curvature Ric M⩾n−1. The paper obtains an inequality for the first eigenvalue η 1 of M with mixed boundary condition, which is a generalization of the results of Lichnerowicz, Reilly, Escobar and Xia. It is also proved that η 1⩾n for certain n-dimensional compact Riemannian manifolds with boundary, which is an extension of the work of Cheng, Li and Yau.

On the extremal functions of Sobolev–Poincaré inequality

hhjWe prove the existence of extremal functions of Sobolev-Poincare inequality on S n for p E (1, (1 + √1+8n)/4). For general n-dimensional compact Riemannian manifolds embedded in R n+1 , such an existence result is proved for p E (n/(n-1), (1 + √1+8n)/4).

SRB measures for Axiom A endomorphisms

Let Λ be a basic set of an Axiom A endomorphism on ndimensional compact Riemannian manifold. In this paper, we provide equivalent conditions for the existence of a SRB measure on Λ. In particular, we show that under the assumption that the closure of the postcritical set of f is disjoint from Λ, the existence of a SRB measure is equivalent to the condition that the stable set of Λ has packing dimension n. Our approach uses the theory of thermodynamic formalism. In particular, we consider the topological pressure on the inverse limit space.

Best constants in Sobolev trace inequalities

In this paper we establish the best constant for a Sobolev trace inequality on compact Riemannian manifolds with boundary. More specifically, let 1< p< n and 1 K ̄ (n,p) = inf ∇u∈L p( R + n) u∈L p ̄ ∗ (∂ R + n)⧹{0} ∫ R + n | ∇u| p ( ∫ ∂ R + n |u| p ̄ ∗ ) p/ p ̄ ∗ , where p ̄ ∗=p(n−1)/(n−p) . We prove that for any compact n-dimensional Riemannian manifold with boundary ( M, g), for any ε>0 there exists A ε >0 such that ||u|| L p ̄ ∗ (∂M) p⩽( K ̄ (n,p)+ε)|| ∇ gu|| L p(M) p+A ε||u|| L p(∂M) p for all u∈ H 1, p ( M).

The Regularity of the Eta Function for Perturbations of Order -(dim X ) of the Atiyah-Patodi-Singer Boundary Problem

Let X be a compact n-dimensional Riemannian manifold with boundary Y and D a Dirac type operator on X. We prove the regularity of the eta function at s = 0 for a (−n)-th order perturbed Atiyah-Patodi-Singer (APS) boundary problem DP . A (−n)-th order perturbed APS boundary condition is determined by a pseudodifferential projection P over the boundary Y which differs from the APS projection by an operator of order −n.

Compact Homogeneous Hypersurfaces in Hyperbolic Space

In this paper we prove that an n-dimensional compact homogeneous Riemannian manifold isometrically immersed in the hyperbolic space Hn+1 is isometric to a sphere.

An inverse problem for a general vibrating annular membrane in [formula omitted] with its physical applications: further results

This paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R 3 . The trace of the heat semigroup θ( t)=∑ v=1 ∞exp(− tμ v ), where { μ v } ∞ v=1 are the eigenvalues of the negative Laplacian −∇ 2=−∑ β=1 3 ∂/ ∂x β 2 in the ( x 1, x 2, x 3)-space, is studied for short-time t for a general vibrating annular membrane Ω in R 3 together with its smooth inner bounding surface S 1 and its smooth outer bounding surface S 2, where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components S i ∗ (i=1,…,m) of S 1 and on the piecewise smooth components S i ∗ (i=m+1,…,n) of S 2 is considered such that S 1=⋃ i=1 mS i ∗ and S 2=⋃ i=m+1 nS i ∗ . In this paper, one may extract information on the geometry of Ω by analyzing the asymptotic expansions of θ( t) for short-time t. Some applications of θ( t) for an ideal gas enclosed in the general vibrating annular membrane Ω are given. We show that the asymptotic expansion of θ( t) for short-time t plays an important role in investigating the influence of the annular region Ω on the thermodynamic quantities of an ideal gas. Some applications of θ( t) for an ideal gas enclosed in a compact n-dimensional Riemannian manifold are also given. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it.

Analysis on Groups of Diffeomorphisms of Manifolds with Boundary and the Averaged Motion of a Fluid

We establish the existence of three new subgroups of the group of volume-preserving diffeomorphisms of a compact n-dimensional (n ≥ 2) Riemannian manifold which are associated with the Dirichlet, Neumann, and Mixed type boundary conditions that arise in second-order elliptic PDEs. We prove that when endowed with the Hs Hilbert-class topologies for s > (n/2) + 1, these subgroups are C∞ differential manifolds. We consider these new diffeomorphism groups with an H1-equivalent right invariant metric, and prove the existence of unique smooth geodesics η(t, ·) of this metric, as well as existence and uniqueness of the Jacobi equations associated to this metric. Geodesics on these subgroups are, in fact, the flows of a time-dependent velocity vector field u(t, x), so that ∂tη(t, ·) = u(t, η(t, ·)) with η(0, x) = x, and remarkably the vector field u(t, x) solves the so-called Lagrangian averaged Euler (LAE-α) equations on M. These equations, and their viscous counterparts, the Lagrangian averaged Navier-Stokes (LANS-α) equations, model the motion of a fluid at scales larger than an a priori fixed parameter α > 0, while averaging (or filtering-out) the small scale motion, and this is achieved without the use of artificial viscosity. We prove that for divergence-free initial data satisfying u = 0 on ∂M, the LAE-α equations are well-posed, globally when n = 2. We also find the boundary conditions that make the LANS-α equations well-posed, globally when n = 3, and prove that solutions of the LANS-α equations converge when n = 2, 3 for almost all t in some fixed time interval (0, T) in Hs, s ∊ (n/2 + 1, 3) to solutions of the LAE-α equations, thus confirming the scaling arguments of Barenblatt & Chorin.

Rigidities of asymptotically Euclidean manifolds

Let M be an n-dimensional compact Riemannian manifold. We study the fundamental group of M when the universal covering space of M, M is close to some Euclidean space ℝs asymptotically.

Compactness of conformal metrics with integral bounds on curvature

In this thesis, we show that a sequence of conformal metrics on a compact n-dimensional Riemannian manifold (n ≥ 4) which has an upper bound on volume and an upper bound on the LP[...] norm of the curvature tensor for fixed p > n/2 has a subsequence which converges in Cα. If n = 3, we have the same result if we assume, in addition, that the scalar curvature has an L2 bound. As corollaries, we have the compactness of a sequence of conformal metrics on a compact three-manifold which are isospectral with respect to either the standard or conformal Laplacian, and the result of Lelong-Ferrand that any compact manifold with non-compact conformal group is conformally equivalent to the standard sphere.

A new algorithm for asymptotic heat kernel expansion for manifolds with boundary

A new algorithm for the computation of the coefficients of the heat kernel expansion, associated with a second-order non-negative elliptic-symmetric differential operator, defined on an N-dimensional compact riemannian manifold with boundary is presented. Some coefficients are explicitly derived. Using zeta function regularization, an explicit form for the expectation value of the matter stress tensor is given. Corrections to a class of anomalies, due to the presence of the boundary, are found.

Metrics of positive scalar curvature on spheres and the Gromov-Lawson conjecture

Boguslaw Hajduk Institute of Mathematics, Wrociaw University, pl. Grunwaldzki 2/4, PL-50-384 Wrodaw, Poland 1. Introduction In [2] and [9] an ingenious procedure is given to construct a Riemannian metric of positive scalar curvature on a manifold obtained by surgery from one which already has such a metric. With some improvements of [1] this may be summarized as follows. 1.1. Theorem. Let (M, g) be a compact n-dimensional Riemannian manifold with positive scalar curvature and let W be a cobordism from M to M' such that W admits a handle decomposition on M with no handles of index greater than n- 2 (i.e. there exists a Morse function on W which is minimal on M and critical points have indices <-_ n- 2). Then there exists a metric of positive scalar curvature on W which extends g and is product on a collar of MUM'. This construction applied to the cobordism with one handle of index 1 between the disjoint sum MuN and the connected sum M4#N shows imme- diately that the connected sum is well defined for manifolds of positive scalar curvature. Furthermore, this operation gives an abelian group structure in the set n~ of concordance classes of positive scalar curvature metrics on S n, with the zero class represented by the standard metric gcan [1]. We say that two metrics go, gl of positive scalar curvature on M are concordant if there exists a metric g of positive scalar curvature on M x [0,1] such that glM x {i} =gi, i=0, 1, and g is product near M x d[0,1]. Our aim is to show how this group, or its subgroup ~ of classes of those metrics which are boundary restrictions of metrics of positive scalar curvature on compact spin manifolds, is related to some questions concerning positive scalar curvature. IfX n is a 2-connected dosed manifold, B a smooth n-baU in X, then by Morse- Srnale theory and Theorem 1.1 there is a metric of positive scalar curvature on X- Int B which is product near S n- i = O(X - Int B). This metric induces a metric of positive scalar curvatures on S "- 1. Our basic observation is that the concordance class 6(X) of the induced metric depends only on the spin cobordism class of X and 6(X)--. 0 if and only if any l-connected manifold spin cobordant to X admits a

Équations de Monge-Ampère invariantes sur les variétés Riemanniennes compactes

Let (Vn, g) be a smooth n-dimensional compact Riemannian manifold (without boundary). Let g′ be a mapping which assigns to the second covariant jet (in the metric g) of any Ck function φ on Vn, k ⩾ 2, a field g′φ twice covariant and symmetric. We take g′ such that there exists φ ∈ Ck(Vn), k ⩾ 2, admissible i.e. for which Trace [(g′φ)−1⋅∂g′φ∂(∇2φ)] is a new metric (everywhere positive definite) (see below and e.g. [3] [4] [5] [6]). Then, given F smooth, one may consider the following non-linear elliptic problem of Monge-Ampère type: find φ ∈ C∞(Vn) admissible, solution of the equation,M(φ)≡(|g′φ|⋅|g|−1)=exp[F(P,∇φ;φ)],where P denotes a generic point of Vn (the admissibility of φ simply means that the symbol of the differential map d[Log M(φ)] is positive definite, hence the ellipticity at φ). In the present article we give existence and uniqueness results for such a Monge-Ampère problem, not in its full generality, but when prescribing on φ→g′φ to be of a form general to be invariant by changes of unknown function of the type φ → ψ, where: ∀P ∈ Vn, ψ(P) ≡ γ[P, φ(P)], γ(P, t) being a function of C∞(Vn × ℝ) such that ∂γ∂t>0.

An eigenvalue pinching theorem

In 1958 Lichnerowicz [7] showed that for a compact n-dimensional riemannian manifold M, whose Ricci curvature is bounded below by n 1 , the first non-zero eigenvalue, 21, of the laplacian satisfies 2 t>n . If, in fact, 21 =n Obata proved that M must be isometric to the standard sphere. A natural question is: Do there exist constants C(n)> 1, depending only on n such that if M is as above and C(n). n> 21 > n then M must be diffeomorphic to a sphere. Here, by combining the works of Gromov [3], Berard and Meyer [1], and Grove and Shiohama [4], we show:

Pinching theorem for the first eigenvalue on positively curved four-manifolds

Let M be an n-dimensional compact Riemannian manifold with Ricci Curvature bounded below by n 1 . A theorem of Lichnerowicz [-4] stated that the first non-zero eigenvalue 21 of the Laplacian acting on functions of M satisfies 21 > n. Obata later proved that equality holds iff M is isometric to the s tandard unit sphere S". One natural question arises is: Does there exists a constant C(n) > 0 depending only on n, such that if in addit ion to the curvature assumption we also have 2 l <n + C(n),

Inequalities of Gauss-Bonnet type for a convex domain

Let N N be a compact convex n n -dimensional Riemannian manifold with a boundary ∂ N \partial N having normal curvatures ⩾ κ &gt; 0 \geqslant \kappa &gt; 0 . Suppose the sectional curvature &gt; − κ 2 &gt; - {\kappa ^2} in N N . Let H H be the integral mean curvature of ∂ N \partial N , V V be the volume of N N , k s c {k_{sc}} be the scalar curvature and k ¯ R ( p ) {\bar k_R}(p) , p ∈ N p \in N , be the maximum Ricci curvature at p p . Then \[ H ⩾ n − 2 2 κ 2 V − 1 2 ( n − 1 ) ∫ N k s c d V , H ⩾ ( n − 2 ) κ 2 V − 1 n − 1 ∫ N k ¯ R d V . H \geqslant \frac {{n - 2}} {2}{\kappa ^2}V - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;dV} ,\quad H \geqslant (n - 2){\kappa ^2}V - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV.} \] Let N − {N_ - } denote N N with nonpositive sectional curvature. Let G G be the integral Gauss curvature of ∂ N − \partial {N_ - } . Then G ⩾ − κ n − 2 ∫ N − k ¯ R d V G \geqslant - {\kappa ^{n - 2}}\int _{N - } {{{\bar k}_R}\;dV} . These three estimates are sharp. For a ball in 3 3 -dimensional hyperbolic space, the ratio of the right-hand part of each estimate to its left-hand part (i.e. V ( κ 2 + 3 ) / 2 H V({\kappa ^2} + 3)/2H , V ( κ 2 + 1 ) / H V({\kappa ^2} + 1)/H and 2 κ V / G 2\kappa V/G respectively) approaches 1 as the radius → ∞ {\operatorname {radius}} \to \infty . The same ratios for the estimates \[ H ⩾ − 1 2 ( n − 1 ) ∫ N k s c d V and H ⩾ − 1 n − 1 ∫ N k ¯ R d V H \geqslant - \frac {1} {{2(n - 1)}}\int _N {{k_{sc}}\;} dV\quad {\text {and}}\quad H \geqslant - \frac {1} {{n - 1}}\int _N {{{\bar k}_R}\;dV} \] (rougher ones but without κ \kappa ) approach 3 4 \tfrac {3} {4} and 1 2 \tfrac {1} {2} respectively.