Thermostats without conjugate points

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

We give a new proof of the existence of the Green bundles. Let (M, g) be a compact Riemannian manifold, and denote by 9t its geodesic flow on the tangent bundle TM. Let -r : TM -* M be the canonical projection and for all 0 E TM let V(0) be the kernel of d-ro. Two points 01, 02 are said to be conjugate if 02 g9tO and dgtV(01) n V(02) & o0 It was proved by Hopf [5] that a two-dimensional torus without conjugate points is flat. Afterwards, Green [8] proved that the integral of the scalar curvature of a manifold without conjugate points is nonpositive and it vanishes if the metric is locally flat. A main ingredient was the existence, under the condition of no conjugate points, of the following bundles: (1) Es(0) lim dg_tV(gt(0)), t-> oo (2) E'(0) = lim dgtV(g_t(0)). t-* oo Hopf's result was generalized to higher dimensions in [2], but there are still new rigidity type results using these bundles; see for example [1]. These bundles have other applications: among other ideas they where used by Freire and Mane' [7] to obtain estimates of the topological entropy. Foulon [6] generalized this result to the case of Finsler metrics. The bundles were also used by Eberlain [4] who proved that these are transverse if and only if the geodesic flow is Anosov. This result was also generalized to the case of convex Hamiltonians without conjugate points; see [31. The purpose of this note is to give a new proof of the following Theorem. If the geodesic flow 9t of a compact manifold does not have conjugate points, then for every 0 in TM the limits (1) and (2) exist. We recall from [4] the definition of the connection map K: T0TM -> T7r(o)M. For ( on T0TM let Z: (-c, e) -* TM be a curve with initial velocity (. Define K(s) = Z'(O) to be the covariant derivative of Z along the curve ir o Z. The definition does not depend on the curve Z. Received by the editors February 8, 2001. 2000 Mathematics Subject Classification. Primary 37D40. The author was partially supported by CONACYT-Mexico grant #28489-E and EPSRCUnited Kingdom GR/M5610. ?)2002 American Mathematical Society

In this paper we show that manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, Section I.5]. The main theorem is that if $X$ is a complete, simply connected Riemannian manifold without conjugate points, and $M=X\times R$ is given the Riemannian product metric $g$, then any metric without conjugate points on $M$ which agrees with $g$ outside a compact set is isometric to $g$.

In this work we prove that the geodesic flow of a compact, n-dimensional Finsler manifold without conjugate points and which is an uniform visibility manifold is transitive. For this, we introduce Finsler versions of Gromov's hyperbolicity and Eberlein's visibility concepts and study its consequences. As an application of the transitivity, we prove that compact, k-basic Finsler surfaces without conjugate points, with genus greater than one and with continuous Green bundles are Riemannian.

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

Using the notion of magnetic curvature recently introduced by the first author, we extend E. Hopf’s theorem to the setting of magnetic systems. Namely, we prove that if the magnetic flow on the s-sphere bundle is without conjugate points, then the total magnetic curvature is non-positive, and vanishes if and only if the magnetic system is magnetically flat. We then prove that magnetic flatness is a rigid condition, in the sense that it only occurs when either the magnetic form is trivial and the metric is flat, or when the magnetic system is Kähler, the metric has constant negative sectional holomorphic curvature, and s equals the Mañé critical value.

We show that the horocyclic flow of an orientable compact higher genus surface without conjugate points and with continuous Green bundles is uniquely ergodic. In particular, the result applies to nonflat nonpositively curved surfaces.

We study the geodesic flow of a compact surface without conjugate points and genus greater than one and continuous Green bundles. Identifying each strip of bi-asymptotic geodesics induces an equivalence relation on the unit tangent bundle. Its quotient space is shown to carry the structure of a 3-dimensional compact manifold. This manifold carries a canonically defined continuous flow which is expansive, time-preserving semi-conjugate to the geodesic flow, and has a local product structure. An essential step towards the proof of these properties is to study regularity properties of the horospherical foliations and to show that they are indeed tangent to the Green subbundles. As an application it is shown that the geodesic flow has a unique measure of maximal entropy.

A conjecture of Burns and Knieper (1991) asks whether a 2-plane with a metric without conjugate points, and with a geodesic foliation whose lines are at bounded Hausdorff distance, is necessarily flat. We prove this conjecture in two cases: under the hypothesis that the plane admits total curvature, and under the hypothesis of visibility at some point. Along the way, we show that all geodesic line foliations on a Riemannian 2-plane must be homeomorphic to the standard one.

We study a complete noncompact minimal submanifold Mn in a sphere Sn+p. We prove there is no nontrivial L2 harmonic 1-form and at most one nonparabolic end on M if the total curvature is bounded from above by a constant depending only on n. The rigidity theorem is a generalized version of Ni’s, Yun’s and the second author’s results on submanifolds in Euclidean spaces and Seo’s result on minimal submanifolds in hyperbolic spaces.

Without any symmetry assumptions and under natural integrability conditions on the Gaussian curvature, we show that the Euclidean plane is unique among all the complete 2-dimensional Riemannian manifolds that satisfy the Euclid’s fifth postulate. Namely, we prove that the Euclidean plane is the only Riemannian surface free of conjugate points and admitting total curvature that satisfies Playfair’s version of the parallel postulate.

A piecewise flat manifold is a triangulated manifold given a geometry by specifying edge lengths (lengths of 1-simplices) and specifying that all simplices are Euclidean. We consider the variation of angles of piecewise flat manifolds as the geometry varies in a particular way, which we call a conformal variation. This variation generalizes variations within the class of circles with fixed intersection angles (such as circle packings) as well as other formulations of conformal variation of piecewise flat manifolds previously suggested. We describe the angle derivatives of the angles in two- and three-dimensional piecewise flat manifolds, giving rise to formulas for the derivatives of curvatures. The formulas for derivatives of curvature resemble the formulas for the change of scalar curvature under a conformal variation of Riemannian metric. They allow us to explicitly describe the variation of certain curvature functionals, including Regge’s formulation of the Einstein-Hilbert functional (total scalar curvature), and to consider convexity of these functionals. They also allow us to prove rigidity theorems for certain analogues of constant curvature and Einstein manifolds in the piecewise flat setting.

We give characterizations of affine transformations and affine vector fields in terms of the spray. By utilizing the Jacobi type equation that characterizes affine vector fields, we prove some rigidity theorems of affine vector fields on compact or forward complete non-compact Finsler manifolds with non-positive total Ricci curvature.

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb {R}^3$ can be approximated uniformly on compacts in $\mathring M=M\setminus bM$ by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb {R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb {R}^n$ for any $n\ge 3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense.

We describe first the analytic structure of Riemann's examples of singly-periodic minimal surfaces; we also characterize them as extensions of minimal annuli bounded by parallel straight lines between parallel planes. We then prove their uniqueness as solutions of the perturbed problem of a punctured annulus, and we present standard methods for determining finite total curvature periodic minimal surfaces and solving the period problems.