The length of a shortest geodesic net on a closed Riemannian manifold

In this paper we will estimate the smallest length of a minimal geodesic net on an arbitrary closed Riemannian manifold Mⁿ in terms of the diameter of this manifold and its dimension. Minimal geodesic nets are critical points of the length functional on the space of immersed graphs into a Riemannian manifold. We prove that there exists a minimal geodesic net that consists of m geodesics connecting two points p,q∈Mⁿ of total length ≤md, where m∈{2,…,(n+1)} and d is the diameter of Mⁿ. We also show that there exists a minimal geodesic net with at most n+1 vertices and geodesic segments of total length .These results significantly improve one of the results of [A. Nabutovsky, R. Rotman, The minimal length of a closed geodesic net on a Riemannian manifold with a nontrivial second homology group, Geom. Dedicata 113 (2005) 234-254] as well as most of the results of [A. Nabutovsky, R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, Geom. Funct. Anal. 14 (4) (2004) 748-790].     

Flowers on Riemannian manifolds

In this paper, we will present two upper bounds for the length of a smallest “flower-shaped” geodesic net in terms of the volume and the diameter of a manifold. Minimal geodesic nets are critical points of the length functional on the space of graphs immersed into a Riemannian manifold. Let M ⁿ be a closed Riemannian manifold of dimension n. We prove that there exists a minimal geodesic net that consists of one vertex and at most 2n ? 1 geodesic loops based at that vertex of total length ≤ 2n!d, where d is the diameter of M ⁿ . We also show that there exists a minimal geodesic net that consists of one vertex and at most ${3^{(n+1)^2}}$ loops of total length ${\leq2 (n+1)!^2 3^{(n+1)^3}\,Fill\,Rad\,M^n \leq2(n+1)!^{\frac{5}{2}}3^{(n+1)^3}(n+1)n^n vol(M^n)^{\frac{1}{n}}}$ , where Fill Rad M ⁿ denotes the filling radius and vol(M ⁿ ) denotes the volume of M ⁿ .     

Geodesic loops and periodic geodesics on a Riemannian manifold diffeomorphic to <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

Let M ⁿ be a closed 2-connected Riemannian manifold, such that π₃(M ⁿ ) ≠ { 0 }. In this paper we prove that either there exists a periodic geodesic on M ⁿ of length ≤ 6d, where d is the diameter of M ⁿ , or at each point p ∈ M ⁿ there exists a geodesic loop of length ≤ 2d.     

Finite Topological Type and Volume Growth

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature Ric _M –(n – 1) and conjugateradius conj _M c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space H ⁿ (–1)of sectional curvature –1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.     

Topological rigidity theorems for open Riemannian manifolds

In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?ⁿ if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?ⁿ if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A metric on the manifold of immersions and its Riemannian curvature

E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n–1) dimensional manifold into ⁿ , and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.     

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and , then for every n > 0, . Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S ¹, if γ₀ is a ramp, then we have a global flow which converges to a closed geodesic in C ^∞ norm. As an application, we prove the theorem of Lyusternik and Fet.      

On the measures of maximal entropy for the geodesic flow

Let M be a closed and connected manifold equipped with a C^∞ Riemannian metric. Using the geodesic arc formalism developed by Maé (J. Differential Geom. 45 (1997) 74–93) and Paternain [Ann. Sci. École Norm. Sup. 4eme série t 33 (2000) 121–138], we give a way of constructing a measure of maximal entropy for the geodesic flow.     

On the period spectrum of a symplectic mapping

Let M be a compact Riemannian manifold. It has been known for a long time that the singularities of the wave trace, trace(cos √Δt), are located at the periods of the closed geodesics. Do these singularities also contain information about the geometry of M in the neighborhood of a closed geodesic? We prove that the Birkhoff canonical form of the Poincaré map can be determined from the singularities of the wave trace.     

Volume,diameter and the minimal mass of a stationary 1-cycle

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold Mⁿ is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold Mⁿ. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold Mⁿ is bounded from above by where where is the filling radius of a manifold, and where is its volume.     

Short Geodesic Segments between Two Points on a Closed Riemannian Manifold

Let M ⁿ be a closed Riemannian manifold homotopy equivalent to the product of S ² and an arbitrary (n–2)-dimensional manifold. In this paper we prove that given an arbitrary pair of points on M ⁿ there exist at least k distinct geodesics of length at most 20k!d between these points for every positive integer k. Here d denotes the diameter of M ⁿ .     

Curvature-free upper bounds for the smallest area of a minimal surface

In this paper we will present two upper estimates for the smallest area of a possibly singular minimal surface in a closed Riemannian manifold Mⁿ with a trivial first homology group. The first upper bound will be in terms of the diameter of Mⁿ, the second estimate will be in terms of the filling radius of a manifold, leading also to the estimate in terms of the volume of Mⁿ. If n = 3 our upper bounds are for the smallest area of a smooth embedded minimal surface. After that we will establish similar upper bounds for the smallest volume of a stationary k-dimensional integral varifold in a closed Riemannian manifold Mⁿ with . The above results are the first results of such nature. Received: October 2004 Revision: May 2005 Accepted: June 2005     

Fundamental group and contractible closed geodesics

We prove the existence of a nonempty class of finitely presented groups with the following property: If the fundamental group of a compact Riemannian manifold M belongs to this class, then there exists a constant c(M) > 1 such that for any sufficiently large x the number of contractible closed geodesics on M of length not exceeding x is greater than c(M)^x. In order to prove this result, we give a lower bound for the number of contractible closed geodesics of length ≤ x on a compact Riemannian manifold M in terms of the resource-bounded Kolmogorov complexity of the word problem for π₁ (M), thus answering a question posed by Gromov. © 1996 John Wiley & Sons, Inc.     

Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K _M 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with K _M 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.     

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT ², we show that the set of Riemannian metrics inT ² with no geodesic foliations having rotation numberh isC ^k dense for everyk N. We also show that, generically in theC ² topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.Partially supported by CNPq-GMD, FAPERJ, and the University of Freiburg.     

Rigidity and sphere theorem for manifolds with positive Ricci curvature

LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China     

Compact minimal hypersurfaces with index one in the real projective space

Let M ⁿ be a compact (two-sided) minimal hypersurface in a Riemannian manifold . It is a simple fact that if has positive Ricci curvature then M cannot be stable (i.e. its Jacobi operator L has index at least one). If is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator.?We prove that if is the real projective space , obtained as a metric quotient of the unit sphere, and the Jacobi operator of M has index one, then M is either a totally geodesic sphere or the quotient to the projective space of the hypersurface obtained as the product of two spheres of dimensions n ₁, n ₂ and radius R ₁, R ₂, with and . Received: June 6, 1998     

The Problem of Prescribed Critical Functions

Let (M,g) be a compact Riemannian manifold on dimension n ≥ 4 not conformally diffeomorphic to the sphere Sⁿ. We prove that a smooth function f on M is a critical function for a metric g conformal to g if and only if there exists x ∈ M such that f(x) > 0.Mathematics Subject Classifications (2000): 53C21, 46E35, 26D10.     

Stability of submanifolds with parallel mean curvature in calibrated manifolds

On a Riemannian manifold $ \bar M^{m + n} $ \bar M^{m + n} with an (m + 1)-calibration Ω, we prove that an m-submanifold M with constant mean curvature H and calibrated extended tangent space ℝH ⋇ TM is a critical point of the area functional for variations that preserve the enclosed Ω-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when n = 1 and Ω is the volume element of $ \bar M $ \bar M . To the second variation we associate an Ω-Jacobi operator and define Ω-stability. Under natural conditions, we show that the Euclidean m-spheres are the unique Ω-stable submanifolds of ℝ ^m+n . We study the Ω-stability of geodesic m-spheres of a fibred space form M ^m+n with totally geodesic (m + 1)-dimensional fibres.     

Kähler hyperbolicity and the Euler number of a compact manifold with geodesic flow of Anosov type

 Let M be a 2m-dimensional compact Riemannian manifold with Anosov geodesic flow. We prove that every closed bounded k form, k≥2, on the universal covering of M is d(bounded). Further, if M is homotopy equivalent to a compact K?hler manifold, then its Euler number χ(M) satisfies (−1) ^m χ(M)>0. Received: 25 September 2001 / Published Online: 16 October 2002