We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn（n 5） is an odd-dimensional compact submanifold with parallel mean curvature in Sn＋p,and if RicM ＆gt;（n- 2-1n）（1 ＋ H2...We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn（n 5） is an odd-dimensional compact submanifold with parallel mean curvature in Sn＋p,and if RicM ＆gt;（n- 2-1n）（1 ＋ H2） and H ＆lt; δn,where δn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn（n 5） is an odd-dimensional compact submanifold in the space form Fn＋p（c） with c 0,and if RicM ＆gt;（n-2-εn）（c＋H2）,where εn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.展开更多
Disorder and localization have dramatic influence on the topological properties of a quantum system.While strong disorder can close the band gap thus depriving topological materials of topological features,disorder ma...Disorder and localization have dramatic influence on the topological properties of a quantum system.While strong disorder can close the band gap thus depriving topological materials of topological features,disorder may also induce topology from trivial band structures,wherein topological invariants are shared by completely localized states.Here we experimentally investigate a fundamentally distinct scenario where topology is identified in a critically localized regime,with eigenstates neither fully extended nor completely localized.Adopting the technique of momentum-lattice engineering for ultracold atoms,we implement a one-dimensional,generalized Aubry-Andrémodel with both diagonal and off-diagonal quasi-periodic disorder in momentum space,and characterize its localization and topological properties through dynamic observables.We then demonstrate the impact of interactions on the critically localized topological state,as a first experimental endeavor toward the clarification of many-body critical phase,the critical analogue of the many-body localized state.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos.11071211,11371315 and 11301476)the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of Chinathe China Postdoctoral Science Foundation (Grant No.2012M521156)
文摘We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn（n 5） is an odd-dimensional compact submanifold with parallel mean curvature in Sn＋p,and if RicM ＆gt;（n- 2-1n）（1 ＋ H2） and H ＆lt; δn,where δn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn（n 5） is an odd-dimensional compact submanifold in the space form Fn＋p（c） with c 0,and if RicM ＆gt;（n-2-εn）（c＋H2）,where εn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.
基金the National Key Research and Development Program of China(2018YFA0307200,2016YFA0301700 and 2017YFA0304100)the National Natural Science Foundation of China(12074337 and 11974331)+2 种基金Natural Science Foundation of Zhejiang Province(LR21A040002 and LZ18A040001)Zhejiang Provincial Plan for Science and Technology(2020C01019)the Fundamental Research Funds for the Central Universities(2020XZZX002-05 and 2021FZZX001-02)。
文摘Disorder and localization have dramatic influence on the topological properties of a quantum system.While strong disorder can close the band gap thus depriving topological materials of topological features,disorder may also induce topology from trivial band structures,wherein topological invariants are shared by completely localized states.Here we experimentally investigate a fundamentally distinct scenario where topology is identified in a critically localized regime,with eigenstates neither fully extended nor completely localized.Adopting the technique of momentum-lattice engineering for ultracold atoms,we implement a one-dimensional,generalized Aubry-Andrémodel with both diagonal and off-diagonal quasi-periodic disorder in momentum space,and characterize its localization and topological properties through dynamic observables.We then demonstrate the impact of interactions on the critically localized topological state,as a first experimental endeavor toward the clarification of many-body critical phase,the critical analogue of the many-body localized state.
