The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localizati...The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the analytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.展开更多
In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nif...In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.展开更多
This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It...This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It is demonstrated that the answer to this question is no, by constructing a pair of disjoint isospectral planar domains whose boundaries have the same interior Bouligand-Minkowski dimension but are not isometric. In the second part of this paper the authors give the exact two-term asymptotics for the Dirichlet counting functions associated with the examples given here and obtain sharp two sided estimates for the second term of the counting functions. The first result in the second part of the paper shows that the coefficient of the second term is an oscillatory function of λ, which implies that the Weyl-Berry conjecture, for the examples given here, is false. The second result implies that the weaker form of the Weyl-Berry conjecture, for these examples, is true. This in turn means that the interior Bouligand-Minkowski dimension of the examples is a spectral invariant.展开更多
基金Project supported by the Cheung-Kong Scholarshipthe Key Laboratory of Pure MathematicsCombinatorics of the Ministry of Education of Chinathe 973 Project of the Ministry of Science and Technology of China.
文摘The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a consequence, it is shown that the R/Z part of the analytically defined η invariant of Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold can be expressed purely geometrically through a stable Chern-Simons current on a higher dimensional sphere. As a preliminary application, the author discusses the relation with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles, and proves an R refinement in the case where the Dirac operator is replaced by the Signature operator.
基金Project supported by the National Natural Science Foundation of China(Nos.10971055,11171096)the Research Fund for the Doctoral Program of Higher Education of China(No.20104208110002)the Funds for Disciplines Leaders of Wuhan(No.Z201051730002)
文摘In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.
文摘This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It is demonstrated that the answer to this question is no, by constructing a pair of disjoint isospectral planar domains whose boundaries have the same interior Bouligand-Minkowski dimension but are not isometric. In the second part of this paper the authors give the exact two-term asymptotics for the Dirichlet counting functions associated with the examples given here and obtain sharp two sided estimates for the second term of the counting functions. The first result in the second part of the paper shows that the coefficient of the second term is an oscillatory function of λ, which implies that the Weyl-Berry conjecture, for the examples given here, is false. The second result implies that the weaker form of the Weyl-Berry conjecture, for these examples, is true. This in turn means that the interior Bouligand-Minkowski dimension of the examples is a spectral invariant.
