A parabolic equation on a manifold of nonpositive curvature

For a fundamental solution of a parabolic equation on an n-dimensional Riemannian manifold we obtain two-sided estimates in terms of the curvature of the manifold.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 4, 1997, pp. 66–69.     

Estimates of derivatives of the heat kernel on a compact Riemannian manifold

We give global estimates on the covariant derivatives of the heat kernel on a compact Riemannian manifold on a fixed finite time interval. The proof is based on analyzing the behavior of the heat kernel along Riemannian Brownian bridge.

     

Covariant derivatives of Jacobi fields on a manifold of nonpositive curvature

We obtain estimates of the covariant derivatives of Jacobi fields along a geodesic on a Riemannian manifold of negative curvature. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 755–764, June, 1998.     

Uniform approximation of the heat kernel on a manifold

We approximate the heat kernel h(x, y, t) on a compact connected Riemannian manifold M without boundary uniformly in \((x,y,t)\in M\times M\times [a,b]\), \(a>0\), by n-fold integrals over \(M^n\) of the densities of Brownian bridges. Moreover, we provide an estimate for the uniform convergence rate. As an immediate corollary, we get a uniform approximation of solutions of the Cauchy problem for the heat equation on M.     

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 . If 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.Received: 27 March 2003, Accepted: 5 April 2004, Published online: 16 July 2004Mathematics Subject Classification (2000): 58E20, 58J35     

Orbihedra of nonpositive curvature

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. Partially supported by MSRI, SFB256 and University of Maryland. Partially supported by MSRI, SFB256 and NSF DMS-9104134.     

On the average curvature of a convex curve in a surface of nonpositive Gaussian curvature

In this paper, the upper bound of the average curvature of a convex curve in a simply connected surface of nonpositive Gaussian curvature is obtained.

     

Foliations with leaves of nonpositive curvature

We answer a question of Gromov ([G2]) in the codimension 1 case: ifF is a codimension 1 foliation of a compact manifoldM with leaves of negative curvature, thenπ ₁(M) has exponential growth. We also prove a result analogous to Zimmer’s ([Z2]): ifF is a codimension 1 foliation on a compact manifold with leaves of nonpositive curvature, and ifπ ₁(M) has subexponential growth, then almost every leaf is flat. We give a foliated version of the Hopf theorem on surfaces without conjugate points. Partially supported by NSF Grant #DMS 9403870.     

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.      

Spaces of nonpositive curvature arising from a finite algebra

In this paper we introduce a family of examples that can be regarded as spaces of nonpositive curvature, but with the distinct quality that they are not complete as metric spaces. This amounts to the fact that they are modelled on a finite von Neumann algebra, and the metrics introduced arise from the trace of the algebra. In spite of the noncompleteness of these manifolds, their geometry can be studied from the view-point of metric geometry, and several techniques derived from the functional analysis are applied to gain insight on their geodesic structure.     

A nonpositive curvature property of modular semilattices

Geometriae Dedicata - The orthoscheme complex of a graded poset is a metrization of its order complex such that the simplex of each maximal chain is isometric to the Euclidean simplex of vertices...     

Riemannian Hilbert submanifolds of nonpositive extrinsic curvature

We consider strongly parabolic, Hilbert submanifolds in Riemannian Hilbert manifolds. We prove that their properties are analogous to the known properties in the finite-dimensional case. The main geometric result consists of Theorem 3: a complete, Riemannian, Hilbert submanifold of nonpositive extrinsic curvature and finite codimension in a Hilbert sphere is a great sphere.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 91–100, 1990.The author expresses his thanks to Professor A. A. Borisenko for posing the problem and guiding the work.     

Manifolds of nonpositive curvature and their buildings

Let M be a complete Riemannian manifold of bounded nonpositive sectional curvature and finite volume. We construct a topological Tits building Δ associated to the universal cover of M. If M is irreducible and rank (M)≥2, we show that Δ is a building canonically associated with a Lie group and hence that M is locally symmetric. Supported in part by NSF Grant MCS-82-04024 and M.S.R.I., Berkeley. Supported in part by NSF Grant DMS-84-01760 and M.S.R.I., Berkeley.     

Bounded harmonic functions on Riemannian manifolds of nonpositive curvature

Certain general conditions are put forth on a complete simply-connected Riemannian manifold of nonpositive curvature which guarantee that they support nontrivial bounded harmonic functions. This result includes the Cartan–Hadamard manifolds with curvature pinched between two negative constants and the bounded symmetric domains \mathfrakR_I(n,n){{\mathfrak{R}}_I(n,n)} and \mathfrakR_II(n){{\mathfrak{R}}_{II}(n)} (n ≥ 2) as special cases.     

Closed geodesics on orbifolds of nonpositive or nonnegative curvature

In this note we prove existence of closed geodesics of positive length on compact developable orbifolds of nonpositive or nonnegative curvature. We also include a geometric proof of existence of closed geodesics whenever the orbifold fundamental group contains a hyperbolic element and therefore reduce the existence problem to developable orbifolds with \(\pi _1^{orb}\) infinite and having finite exponent and finitely many conjugacy classes.     

Eigenvalue inequalities for the p-Laplacian on a Riemannian manifold and estimates for the heat kernel

In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p  -Laplacian (1<p<∞$1<p<\infty$) obtained by Matei (2000) [19] and Takeuchi (1998) [22], respectively. Moreover, we use this generalized eigenvalue comparison theorem to get estimates for the first eigenvalue of the Dirichlet p-Laplacian of geodesic balls on complete Riemannian manifolds with radial Ricci curvature bounded from below w.r.t. some point. In the rest of this paper, we derive an upper and lower bound for the heat kernel of geodesic balls of complete manifolds with specified curvature constraints, which can supply new ways to prove the most part of two generalized eigenvalue comparison results given by Freitas, Mao and Salavessa (2013) [9].