Almost periodic measures on the torus

Given a skew product flow (T,T ²) on the two torus, we construct a family of flows onT ³ parametrized by elements of the circleT. We show that under a certain condition on (T,T ²) almost every flow in this family is strictly ergodic. This is used to characterize minimal subsets of the flow (T,P(T ²)) induced byT on the space of probability measures onT ². Using a result of M. Herman, we give an example to show that this characterization does not hold for everyT. To the memory of Shlomo Horowitz     

L 2 Forms and Ricci Flow with Bounded Curvature on Complete Non-Compact Manifolds

In this paper, we study the evolution of L ² one forms under Ricci flow with bounded curvature on a non-compact Rimennian manifold. We show on such a manifold that the L ² norm of a smooth one form is non-increasing along the Ricci flow with bounded curvature. The L ^∞ norm is showed to have monotonicity property too. Then we use L ^∞ cohomology of one forms with compact support to study the singularity model for the Ricci flow on .     

Heat flow on Finsler manifolds

This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : T_xM → ?₊ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold:

• as gradient flow on L²(M, m) for the energy
• as gradient flow on the reverse L²‐Wasserstein space ??₂(M) of probability measures on M for the relative entropy

Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove ??^{1, α} regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be ??². Moreover, we derive pointwise comparison results à la Cheeger‐Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc.     

Unique ergodicity of flows on the torus and the rotation number

We prove in this paper that any flow on the 2- torus with no singular points and periodic orbits which is generated by a vector fieldV=(P, Q) satisfyingV∈C ¹ orV∈C ⁰ andP≠0 is uniquely ergodic. Then we give an expression of the rotation number by using an invariant measure of a flow.     

On pulsatile flow of a viscous fluid in a rotating channel

A study is made on the pulsatile flow superposed on a steady laminar flow of a viscous fluid in a parallel plate channel rotating with an angular velocity Ω about an axis perpendicular to the plates. An exact solution of the governing equations of motion is obtained. The solution in dimensionless form contain two parametersK ²=ΩL ²/v which is reciprocal of Ekmann Number and frequency parameter σ=αL ²/v. The effects of these parameters on the principal flow characters such as mean sectional velocity and shear stresses at the plates have been examined. For large σ andK ² the flow near the plates has a multiple boundary layer character.     

Examples for cross curvature flow on 3-manifolds

Recently, B. Chow and R.S. Hamilton [3] introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over a circle, and the other is on a S² bundle over a circle. We show that the global flow exists in both cases. However, on the former the flow diverges at time infinity, and on the latter the flow converges at time infinity. Mathematics Subject Classification (1991) 53C44     

On Morse Conjecture for Flows on Closed Surfaces

Let M be a closed orientable surface and let ϕ be a C¹‐flow on M with set of singularities compact countable. In this paper, we prove the Morse conjecture for ϕ: if ϕ is topologically transitive then it is metrically transitive.     

Energy of Flows on Percolation Clusters

It is well known for which gauge functions H there exists a flow in Z ^d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z ^d we let our (random) graph cal C cal (Z ^d,p) be the graph obtained from Z ^d by removing edges with probability 1–p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d3,p>p _c(Z ^d), simple random walk on cal C cal (Z ^d,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x ² energy _e f(e)² is finite. Levin and Peres (1998) sharpened this result, and showed that if d3 and p>p _c(Z ^d), then cal C cal (Z ^d,p) supports a nonzero flow f such that the x ^q energy is finite for all q>d/(d–1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z ^d. In this paper we close the gap by showing that if d3 and Z ^d supports a flow of finite H energy then the infinite percolation cluster on Z ^d also support flows of finite H energy. This disproves a conjecture of Levin and Peres.     

Characterization of Geodesic Flows on T 2 with and without Positive Topological Entropy

In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T ² for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \mathbbR²{\mathbb{R}^{2}} . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T ².     

The Hyperbolic Geometric Flow on Riemann Surfaces

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ?² in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric g _ij keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably “large", then the solution metric g _ij converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature R(t, x) goes to positive infinity as (t, x) tends to the blowup points, and a flow with surgery has to be considered. The authors attempt to show that, comparing to Ricci flow, the hyperbolic geometric flow has the following advantage: the surgery technique may be replaced by choosing suitable initial velocity tensor. Some geometric properties of hyperbolic geometric flow on general open and closed Riemann surfaces are also discussed.     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.     

On Anosov energy levels of hamiltonians on twisted cotangent bundles

LetT ^* M denote the cotangent bundle of a manifoldM endowed with a twisted symplectic structure [1]. We consider the Hamiltonian flow generated (with respect to that symplectic structure) by a convex HamiltonianH: T ^* M, and we consider a compact regular energy level ofH, on which this flow admits a continuous invariant Lagrangian subbundleE. When dimM3, it is known [9] that such energy level projects onto the whole manifoldM, and thatE is transversal to the vertical subbundle. Here we study the case dimM=2, proving that the projection property still holds, while the transversality property may fail. However, we prove that in the case whenE is the stable or unstable subbundle of an Anosov flow, both properties hold.     

Gradient flows with wildly embedded closures of separatrices

We show that for any n ≥ 4 there exists an n-dimensional closed manifold M ⁿ on which one can define a Morse-Smale gradient flow f ^t with two nodes and two saddles such that the closure of the separatrix of some saddle of f ^t is a wildly embedded sphere of codimension 2. We also prove that the closures of separatrices of a flow with three equilibrium points are always embedded in a locally flat way.     

Square function and heat flow estimates on domains

The first purpose of this note is to provide a proof of the usual square function estimate on L^p(Ω). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, and we provide a sketch of its proof in the Appendix for the reader’s convenience. We also relate such bounds to a weaker version of the square function estimate which is enough in most instances involving dispersive PDEs and relies on Gaussian bounds on the heat kernel (such bounds are the key to Alexopoulos’result as well). Moreover, we obtain several useful L^p(Ω;H) bounds for (the derivatives of) the heat flow with values in a given Hilbert space H.     

On the conditions to control curvature tensors of Ricci flow

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that L^\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ  × [0, T), the curvature tensor stays uniformly bounded on M ⁿ  × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.     

A set of global bounded solutions for a Volterra system of retarded equations on \Bbb R3+ {\Bbb R}^3_+

It is proved the existence of a compact set , invariant under the flow of a Volterra system of retarded equations on , with lag r > 0; is homeomorphic to a solid tri-dimensional cylinder. The boundary of is the union of a closed bi-dimensional cylinder with two open disks (the two basis of the cylinder ). is the union of a continuous one-parameter family of r-periodic orbits of the retarded Volterra system and any r-periodic orbit of the retarded system is contained in . The flow, restricted to , of the system of retarded equations, is the flow of a C ¹-vector-field.     

Rate of convergence of the mean curvature flow

We study the flow M_t of a smooth, strictly convex hypersurface by its mean curvature in ?^{n + 1}. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x^* (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere Sⁿ of radius √n. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us that the rate of the exponential decay is at least 2/n. We can define the “arrival time” u of a smooth, strictly convex, n‐dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u(x) = t if x ∈ M_t for x ∈ Int(M₀). Huisken proved that, for n ≥ 2, u(x) is C² near x^*. The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C³‐regularity of u. As a consequence of the obtained rate of convergence of the mean curvature flow, we prove that u is not necessarily C³ near x^* for n ≥ 2. We also show that the obtained rate of convergence 2/n, which arises from linearizing a mean curvature flow, is the optimal one, at least for n ≥ 2. © 2007 Wiley Periodicals, Inc.     

Quick Max-flow Algorithm

Determining of a maximal network flow is a classic problem in discrete optimization with many applications. In this paper, a new algorithm based on the Dinic’s method is presented. Algorithms of the Dinic’s method work evidently faster than theoretical bounds for a randomized network. This paper presents a parameterized and easy to implement family of algorithms of finding a saturating flow in a layered network. Although their common complexity is poor O(V ² L) where L is the number of layers, three particular members are proved to be O(V ²). Furthermore, there is a particularly interesting “balanced” member of the family for which a calculated upper bound on complexity is still O(V ² L) but there is known no example of a layered network that needs more than O(E + V ^(3/2)) time to resolve. All the considered members work really quickly for randomized examples of a layered network. Starting from the above family, three algorithms which find maximal flow in a network in O(V ³) worst case time have been constructed, while the respective “balanced” algorithm is theoretically O(V ⁴). All the algorithms do not extend O(V ²) time in experimental, i.e. randomized, cases.      

Linear,hydrodynamic flow in a rotating saturated porous medium

Linear, steady, axisymmetric flow of a homogeneous fluid in a rigid, bounded, rotating, saturated porous medium is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The dynamics of the interior region and vertical boundary layers are investigated as functions of the Ekman number E(=v/ΩL ²) and rotational Darcy 3 numberN(=kΩ/v) which measures the ratio between the Coriolis force and the Darcy frictional term. IfN≫E ^−1/2, the permeability is sufficiently high and the flow dynamics are the same as those of the conventional free flow problem with Stewartson'sE ^1/3 andE ^1/4 double layer structure. For values ofN≤E ^−1/2 the effect of porous medium is felt by the flow; the Taylor-Proudman constraint is no longer valid. ForN≪E ^−1/3 the porous medium strongly affects the flow; viscous side wall layer is absent to the lowest order and the fluid pumped by the Ekman layer, returns through a region of thicknessO(N ⁻¹). The intermediate rangeE ^−1/3≪N≪E ^−1/2 is characterized by double side wall layer structure: (1)E ^1/3 layer to return the mass flux and (ii) (NE)^1/2 layer to adjust the interior azimuthal velocity to that of the side wall. Spin-up problem is also discussed and it is shown that the steady state is reached quickly in a time scaleO(N).     

Regularity of the Anosov distributions of Euler-Lagrange deformations of geodesic flows

Let g be a negatively curved Riemannian metric of a closed C ^∞ manifold M of dimension at least three. Let L _λ be a C ^∞ one-parameter convex superlinear Lagrangian on TM such that L₀(v) = \frac12 g(v, v){L_0(v)= \frac{1}{2} g(v, v)} for any v ∈ TM. We denote by j^l{\varphi^\lambda} the restriction of the Euler-Lagrange flow of L _λ on the \frac12{\frac{1}{2}} -energy level. If λ is small enough then the flow j^l{\varphi^\lambda} is Anosov. In this paper we study the geometric consequences of different assumptions about the regularity of the Anosov distributions of j^l{\varphi^\lambda} . For example, in the case that the initial Riemannian metric g is real hyperbolic, we prove that for λ small, j^l{\varphi^\lambda} has C ³ weak stable and weak unstable distributions if and only if j^l{\varphi^\lambda} is C ^∞ orbit equivalent to the geodesic flow of g.