Manifolds Of Nonpositive Curvature Research Articles

The Uniqueness of Knieper Measure on Non-compact Rank 1 Manifolds of Non-positive Curvature

The Uniqueness of Knieper Measure on Non-compact Rank 1 Manifolds of Non-positive Curvature

Fenchel Duality Theory and a Primal-Dual Algorithm on Riemannian Manifolds

This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas–Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm may even converge on manifolds of positive curvature.

On Cohomogeneity Two Riemannian Manifolds of Non-Positive Curvature

We study a cohomogeneity two Riemannian G-manifold M of non-positive curvature. Considering the acting group G, we obtain some facts about the structure of such manifolds and their orbits. Moreover in some cases, the existence of the G-invariant metrics with non-positive (or negative) curvature on M is proved.

Distorted Plane Waves on Manifolds of Nonpositive Curvature

We will consider the high frequency behaviour of distorted plane waves on manifolds of nonpositive curvature which are Euclidean or hyperbolic near infinity, under the assumption that the curvature is negative close to the trapped set of the geodesic flow and that the topological pressure associated to half the unstable Jacobian is negative. We obtain a precise expression for distorted plane waves in the high frequency limit, similar to the one in Guillarmou and Naud (Am J Math 136:445–479, 2014) in the case of convex co-compact manifolds. In particular, we will show \({L_{loc}^\infty}\) bounds on distorted plane waves that are uniform with frequency. We will also show a small-scale equidistribution result for the real part of distorted plane waves, which implies sharp bounds for the volume of their nodal sets.

Improved critical eigenfunction estimates on manifolds of nonpositive curvature

We prove new improved endpoint, $L^{p_c}$, $p_c=\tfrac{2(n+1)}{n-1}$, estimates (the kink point) for eigenfunctions on manifolds of nonpositive curvature. We do this by using energy and dispersive estimates for the wave equation as well as new improved $L^p$, $2<p< p_c$, bounds of Blair and the author \cite{BSTop}, \cite{BSK15} and the classical improved sup-norm estimates of Berard~\cite{Berard}. Our proof uses Bourgain's \cite{BKak} proof of weak-type estimates for the Stein-Tomas Fourier restriction theorem \cite{Tomas}--\cite{Tomas2} as a template to be able to obtain improved weak-type $L^{p_c}$ estimates under this geometric assumption. We can then use these estimates and the (local) improved Lorentz space estimates of Bak and Seeger~\cite{BakSeeg} (valid for all manifolds) to obtain our improved estimates for the critical space under the assumption of nonpositive sectional curvatures.

Improved Critical Eigenfunction Restriction Estimates on Riemannian Surfaces with Nonpositive Curvature

We show that one can obtain improved L 4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge’s strategy in (Improved critical eigenfunction estimates on manifolds of nonpositive curvature, Preprint). We first combine the improved L 2 restriction estimate of Blair and Sogge (Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint) and the classical improved $${L^\infty}$$ estimate of Bérard to obtain an improved weak-type L 4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger (Math Res Lett 18(4):767–781, 2011). This estimate improves the L 4 restriction estimate of Burq et al. (Duke Math J 138:445–486, 2007) and Hu (Forum Math 6:1021–1052, 2009) by a power of $${(\log\log\lambda)^{-1}}$$ . Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of $${(\log\lambda)^{-1}}$$ by applying the ideas from (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) and (Blair and Sogge, Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint). We are able to compute various constants that appeared in (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover $${\mathbb{H}^2}$$ .

Asymptotic rank of spaces with bicombings

The question, under what geometric assumptions on a space X an n-quasiflat in X implies the existence of an n-flat therein, has been investigated for a long time. It was settled in the affirmative for Busemann spaces by Kleiner, and for manifolds of non-positive curvature it dates back to Anderson and Schroeder. We generalize the theorem of Kleiner to spaces with bicombings. This structure is a weak notion of non-positive curvature, not requiring the space to be uniquely geodesic. Beside a metric differentiation argument, we employ an elegant barycenter construction due to Es-Sahib and Heinich by means of which we define a Riemannian integral serving us in a sort of convolution operation.

Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of non-positive curvature

Let (M,g) be a complete Riemannian manifold with nonpositive scalar curvature, let ⊂ M be a suitable domain, and let λ() be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on . We prove several bounds for the rate of decrease of λ() as increases, and a result comparing the rate of decrease of λ before and after

Localized Lp-estimates of eigenfunctions: A note on an article of Hezari and Rivière

We use a straightforward variation on a recent argument of Hezari and Rivière [8] to obtain localized Lp-estimates for all exponents larger than or equal to the critical exponent pc=2(n+1)n−1. We are able to do this directly by just using the Lp-bounds for spectral projection operators from our much earlier work [12]. The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic, all of the Lp, 2<p≤∞, bounds of the corresponding eigenfunctions are relatively small compared to the general ones in [12], which are saturated on round spheres. The connection with quantum ergodicity was established for exponents 2<p<pc in the recent results of the author [13] and Blair and the author [3]; however, the article of Hezari and Rivière [8] was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, pc. As is well known, and we indicate here, bounds for the critical exponent, pc, imply ones for all the other exponents 2<p≤∞. The localized estimates involve L2-norms over small geodesic balls Br of radius r, and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature. We shall also state a problem of determining when one can improve on the trivial O(r12) estimates for these L2(Br) bounds. If r=λ−1, one can improve on the trivial estimates if one has improved Lpc(M) bounds just by using Hölder's inequality; however, obtaining improved bounds for r≫λ−1 seems to be subtle.

Nonflatness of certain contact and CR manifolds

We prove that, under a suitable geometric condition, a Riemannian manifold of dimension at least 7 endowed with a contact distribution cannot be flat. This result yields nonflatness of some classes of almost contact metric manifolds, contact sub-Riemannian symmetric spaces, locally symmetric C R spaces and C R submanifolds of Kähler manifolds. As an application, we prove that a compact flat Riemannian manifold of odd dimension at least 7 cannot be isometrically immersed as a hypersurface of a simply connected, complete Kähler manifold of nonpositive curvature.

Harmonic Maps and Biharmonic Maps

This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.

Improvement of eigenfunction estimates on manifolds of nonpositive curvature

Abstract Let (M,g) be a compact, boundaryless manifold of dimension n with the property that either (i) n = 2 and (M,g) has no conjugate points, or (ii) the sectional curvatures of (M,g) are nonpositive. Let Δ be the positive Laplacian on M determined by g. We study the L 2 → Lp mapping properties of a spectral cluster of (Δ)1/2 of width 1/log λ. Under the geometric assumptions above, Bérard [Math. Z. 155 (1977), 249–276] obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a (log λ)1/2 improvement for Hörmander's estimate on the L ∞ norms of eigenfunctions. In this paper we extend this improvement to the Lp estimates for all p &gt; 2(n+1)/(n-1).

Non existence of homogeneous contact metric manifolds of non positive curvature

We prove that there exist no simply connected homogeneous contact metric manifolds having nonpositive sectional curvature.

Fast diffusion flow on manifolds of nonpositive curvature

We consider the fast diffusion equation (FDE) u t = Δu m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L p −L q smoothing effects of the type ∥u(t)∥ q ≤ Ct −α ∥u 0∥γ p , the case q = ∞ being included. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−Δ) > 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.

Jacobi fields on a Riemann manifold

Some properties of Jacobi fields on a manifold of nonpositive curvature are considered. As a result, we obtain relations for derivatives of one class of functions on the manifold.

Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.

Nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature

Nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature

Nonexistence of complete conformal metrics with prescribed scalar curvatures on a manifold of nonpositive curvature

Let ( M, g) be a complete simply-connected Riemannian manifold with nonpositive curvature, k its scalar curvature, and K a smooth function on M. We obtain a nonexistence result of complete metrics on M conformal to g and with K as their scalar curvature.

Biharmonic map heat flow into manifolds of nonpositive curvature

Let M m and N be two compact Riemannian manifolds without boundary. We consider the L 2 gradient flow for the energy $\int_M\vert\Delta u\vert^2$ . If $m \leq 4$ and N has nonpositive sectional curvature we show that the biharmonic map heat flow exists for all time, and that the solution subconverges to a smooth harmonic map as time goes to infinity. This reproves the celebrated theorem of Eells and Sampson [6] on the existence of harmonic maps in homotopy classes for domain manifolds with dimension less than or equal to 4.

Perturbation Method for a Parabolic Equation with Drift on a Riemannian Manifold

We construct a fundamental solution of a parabolic equation with drift on a Riemannian manifold of nonpositive curvature by the perturbation method on the basis of a solution of an equation without drift. We establish conditions for the drift field under which this method is applicable.