Finiteness theorems for submersions and souls

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose and tensors are both bounded in norm.

     

From local to global behavior in competitive Lotka-Volterra systems

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics.

The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point and the carrying simplex of the system lies to one side of its tangent hyperplane at , then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

     

Integral equations, implicit functions, and fixed points

The problem is to show that (1) has a solution, where defines a contraction, , and defines a compact map, . A fixed point of would solve the problem. Such equations arise naturally in the search for a solution of where , but so that the standard conditions of the implicit function theorem fail. Now would be in the form for a classical fixed point theorem of Krasnoselskii if were a contraction. But fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that has enough properties that an extension of Krasnoselskii's theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.

     

An obstruction to the conformal compactification of Riemannian manifolds

In this paper, we study noncompact complete Riemannian -manifolds with which are not pointwise conformal to subdomains of any compact Riemannian -manifold. For this, we compare the Sobolev Quotient at infinity of a noncompact complete Riemannian manifold with that of the singular set in a compact Riemannian manifold using the method for the Yamabe problem.

     

Metric character of Hamilton-Jacobi equations

We deal with the metrics related to Hamilton-Jacobi equations of eikonal type. If no convexity conditions are assumed on the Hamiltonian, these metrics are expressed by an - formula involving certain level sets of the Hamiltonian. In the case where these level sets are star-shaped with respect to 0, we study the induced length metric and show that it coincides with the Finsler metric related to a suitable convexification of the equation.

     

Global weak solutions to Landau-Lifshitz equations into compact Lie algebras

We consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra g; which can be viewed as the extension of Landau-Lifshitz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such LL equations from an n-dimensional closed Riemannian manifold T or a bounded domain in {ℝ}^n into a unit sphere {{\displaystyle S}}_g(\mathrm{1}) in g. In particular, we consider the Hamiltonian system associated with the nonlocal energy-micromagnetic energy defined on a bounded domain of {ℝ}^{\mathrm{3}} and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.     

The Goulden—Jackson cluster method: extensions,applications and implementations

Th powerful (and so far under-utilized) Goulden—Jackson Cluster method for finding the generating function for the number of words avoiding, as factors, the members of a prescribed set of ‘dirty words’, is tutorialized and extended in various directions. The authors' Maple implementations, contained in several Maple packages available from this paper's website www.math.temple.edu/zeilberg/gj.html, ar described and explained.     

Newton's method on Riemannian manifolds: Smale's point estimate theory under the {gamma}-condition

^** Email: cli{at}zju.edu.cn^*** Email: wjh{at}zjut.edu.cn The -conditions for vector fields on Riemannian manifolds areintroduced. The -theory and the -theory for Newton's methodon Riemannian manifolds are established under the -conditions.Applications to analytic vector fields are provided and theresults due to Dedieu et al. (2003, IMA J. Numer. Anal., 23,395–419) are improved.     

Hyperbolic minimizing geodesics

We apply the intersection theory for Lagrangian submanifolds to obtain a Sturm type comparison theorem for linearized Hamiltonian flows. Applications to the theory of geodesics are considered, including a sufficient condition that arclength minimizing closed geodesics, for an -dimensional Riemannian manifold, are hyperbolic under the geodesic flow. This partially answers a conjecture of G. D. Birkhoff.

     

On transversely flat conformal foliations with good measures

Transversely flat conformal foliations with good transverse invariant measures are Riemannian in the sense. In particular, transversely similar foliations with good measures are transversely Riemannian as transversely -foliations.

     

The length of a shortest closed geodesic and the area of a -dimensional sphere

Let be a Riemannian manifold homeomorphic to . The purpose of this paper is to establish the new inequality for the length of a shortest closed geodesic, , in terms of the area of . This result improves previously known inequalities by C.B. Croke (1988), by A. Nabutovsky and the author (2002) and by S. Sabourau (2004).

     

Growth of fundamental groups and isoembolic volume and diameter

Some properties of fundamental groups of Riemannian manifolds will be studied without a lower bound assumption on Ricci curvature. The main method is to relate the local packing to global packing instead of using the Bishop-Gromov relative volume comparison. This method allows us to control the volume growth of the universal cover and yields bounds on the number of generators of in terms of some isoembolic geometric invariants of .

     

Convergence of gauge method for incompressible flow

A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field and a gauge variable , , was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint . We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.

     

Topological entropy for geodesic flows on fibre bundles over rationally hyperbolic manifolds

Let be the total space of a fibre bundle with base a simply connected manifold whose loop space homology grows exponentially for a given coefficient field. Then we show that for any Riemannian metric on , the topological entropy of the geodesic flow of is positive. It follows then, that there exist closed manifolds with arbitrary fundamental group, for which the geodesic flow of any Riemannian metric on has positive topological entropy.

     

A remark on the maximum principle and stochastic completeness

We prove that the stochastic completeness of a Riemannian manifold is equivalent to the validity of a weak form of the Omori-Yau maximum principle. Some geometric applications of this result are also presented.

     

Boundary rigidity and stability for generic simple metrics

We study the boundary rigidity problem for compact Riemannian manifolds with boundary : is the Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function known for all boundary points and ? We prove in this paper local and global uniqueness and stability for the boundary rigidity problem for generic simple metrics. More specifically, we show that there exists a generic set of simple Riemannian metrics such that for any , any two Riemannian metrics in some neighborhood of having the same distance function, must be isometric. Similarly, there is a generic set of pairs of simple metrics with the same property. We also prove Hölder type stability estimates for this problem for metrics which are close to a given one in .     

Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces

In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold , denoted respectively by and , which trivially satisfy . In this article, we completely determine the Riemannian manifolds satisfying the condition . By applying the notions of these -invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.

     

Subsonic irrotational flows in a two-dimensional finitely long curved nozzle

This is a continuation of our previous paper Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf), where we have characterized a set of physical boundary conditions that ensures the existence and uniqueness of subsonic irrotational flow in a flat nozzle. In this paper, we will investigate the influence of the incoming flow angle and the geometry structure of the nozzle walls on subsonic flows in a finitely long curved nozzle. It turns out to be interesting that the incoming flow angle and the angle of inclination of nozzle walls play the same role as the end pressure for the stabilization of subsonic flows. In other words, the L ² and L ^∞ bounds of the derivative of these two quantities cannot be too large, similar as we have indicated in Du et al. (http://www.ims.cuhk.edu.hk/publications/reports/2012-06.pdf) for the end pressure. The curvatures of the nozzle walls will also play an important role in the stability of the subsonic flow.