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Florent Balacheff 《Geometriae Dedicata》2006,121(1):61-71
We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with
a smooth Riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the
space of metrics, admits the standard metric g
0 as a critical point, although it does not achieve the conjectured global minimum: we show that for each tangent direction
to the space of metrics at g
0, there exists a variation by metrics corresponding to this direction along which the systolic area can only increase.
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Local well‐posedness of critical nonlinear Schrödinger equation on Zoll manifolds of odd‐growth
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Tengfei Zhao 《Mathematical Methods in the Applied Sciences》2016,39(12):3226-3242
In this paper, we study the nonlinear Schrödinger equation on Zoll manifolds with odd order nonlinearities. We will obtain the local well‐poesdness in the critical space . This extends the recent results in the literature to the Zoll manifolds of dimension d≥2 with general odd order nonlinearities and also partially improves the previous results in the subcritical spaces of Yang to the critical cases. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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Florent Balacheff Christopher Croke Mikhail G. Katz 《Geometric And Functional Analysis》2009,19(1):1-10
We construct a counterexample to a conjectured inequality L ≤ 2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin’s theorem
concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus
the round metric is not optimal for the ratio L/D.
F.B. supported by the Swiss National Science Foundation. C.C. supported by NSF grants DMS 02-02536 and DMS 07-04145. M.K.
supported by the Israel Science Foundation (grants 84/03 and 1294/06) 相似文献
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Daoyuan Fang Changqing Tong Sijia Zhong 《Journal of Mathematical Analysis and Applications》2007,331(1):21-37
In this paper we prove a global existence result for nonlinear Klein-Gordon equations with small data in infinite homogeneous waveguids, R2×M, where M=(M,g) is a Zoll manifold. The method is based on the normal forms, the eigenfunction expansion for M and the special distribution of eigenvalues of Laplace-Beltrami on Zoll manifold. 相似文献
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