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Register a new userIntegrable geodesic flow with positive topological entropy
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An example of a real-analytic metric on a compact manifold whose geodesic flow is Liouville integrable by $C^infty$ functions and has positive topological entropy is constructed.
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For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic flow, which has vanishing Liouville entropy and, moreover, is integrable but has positive topological entropy. The authors also discuss some open problems on integrability of geodesic flows and related subjects.
In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamiltons classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelmans second paper on the Ricci flow.
We completely integrate the magnetic geodesic flow on a flat two-torus with the magnetic field $F = cos (x) dx wedge dy$ and describe all contractible periodic magnetic geodesics. It is shown that there are no such geodesics for energy $E geq 1/2$, for $E< 1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.
We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.
Inspired by work of Colding-Minicozzi on mean curvature flow, Zhang introduced a notion of entropy stability for harmonic map flow. We build further upon this work in several directions. First we prove the equivalence of entropy stability with a more computationally tractable $mathcal F$-stability. Then, focusing on the case of spherical targets, we prove a general instability result for high-entropy solitons. Finally, we exploit results of Lin-Wang to observe long time existence and convergence results for maps into certain convex domains and how they relate to generic singularities of harmonic map flow.
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