共查询到20条相似文献,搜索用时 51 毫秒
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We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold . We first assume the pull back Weitzenböck operator of bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when is the space form. 相似文献
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In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic . We assume that the critical point lies on the discontinuity surface . We consider 4 scenarios which differ for the presence or not of sliding close to and for the possible presence of a transversal crossing between and . We assume that the systems are subject to a small non-autonomous perturbation, and we obtain 4 new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics. 相似文献
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We deal here with planar analytic systems which are small perturbations of a period annulus. For each transversal section Σ to the unperturbed orbits we denote by the time needed by a perturbed orbit that starts from to return to Σ. We call this the flight return time function. We say that the closed orbit Γ of is a continuable critical orbit in a family of the form if, for any and any Σ that passes through q, there exists a critical point of such that as . In this work we study this new problem of continuability.In particular we prove that a simple critical periodic orbit of is a continuable critical orbit in any family of the form . We also give sufficient conditions for the existence of a continuable critical orbit of an isochronous center . 相似文献
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