Quantitative Volume Space Form Rigidity Under Lower Ricci Curvature Bound

Let $M$ be a compact $n$-manifold of $\operatorname{Ric}_M\ge (n-1)H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $\rho>0$ such that for any $x\in M$, the open $\rho$-ball at $x^*$ in the (local) Riemannian universal covering space, $(U^*_\rho,x^*)\to (B_\rho(x),x)$, has the maximal volume i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H=-1$, the volume entropy of $M$ is maximal i.e. $n-1$ ([LW1]). The main results of this paper are quantitative space form rigidity i.e., statements that $M$ is diffeomorphic and close in the Gromov-Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H=1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].Comment: The only change from the early version is an improvement on Theorem A: we replace the non-collapsing condition on $M$ by on $\tilde M$ (the Riemannian universal cover), and the corresponding modification is adding "subsection c" in Section

Microscopic Study of Structure, Chemical Composition and Local Conductivity of La2/3Sr1/3MnO3 Films

The colossal magnetoresistance (CMR) manganites have attracted intensive study due to their richness of underlying physics and potential technological applications. Of particular interest is half-metallic La2/3Sr1/3MnO3 (LSMO) because it possesses the highest known Curie temperature of the group ~ 370 K), which makes it a promising candidate for room temperature spintronic applications. On the other hand, LSMO ultrathin films exhibit a metal-insulator transition (MIT) when reducing film thickness. The origin of such a thickness-dependent MIT remains highly controversial, though understanding and controlling this kind of behavior is necessary for any possible device applications. An essential first step then, and the objective of this thesis project, is the characterization of the lattice structure and chemical composition. The chemical composition of LSMO films grown on TiO2-terminated SrTiO3 (001) is quantified with unit cell precision by combining in-situ angle-resolved x-ray photoelectron spectroscopy (ARXPS), ex-situ scanning transmission electron microscopy (STEM), and electron energy loss spectroscopy (EELS). Substantial deviations in Sr doping concentrations from its bulk value are observed at both the interface and surface. Deviation at the interface is due mainly to single unit cell intermixing, while in proximity to the surface the segregation occurs in a wider thickness range. The surface undergoes a gradual conversion from MnO2 to (La/Sr)O layer termination with increasing thickness. To study the consequences of the surface Sr segregation, scanning tunneling spectroscopy (STS) is applied to study the local electronic properties. According to the STS results, the nonmetallic character and spontaneous polarization at the surface of both thin and thick LSMO films is revealed. The difference in surface behavior from the bulk is also confirmed by the temperature-dependent X-ray photoemission spectroscopy (XPS). Sr surface concentration deviation from the bulk value is unambiguously related to the nonmetallic behavior at the surface and interface, which is further verified by the thickness dependence of the film conductivity. The layer-by-layer variation in chemical composition generates an immense impact on the physical properties of the epitaxial oxide films and heterostructures. It naturally explains the existence of a \u27dead\u27 layer and the persistent nonmetallic behavior near the surface and interface of LSMO films, regardless their thickness

The Minimal GUT with Inflaton and Dark Matter Unification

Giving up the solutions to the fine-tuning problems, we propose the non-supersymmetric flipped $SU(5)\times U(1)_X$ model based on the minimal particle content principle, which can be constructed from the four-dimensional $SO(10)$ models, five-dimensional orbifold $SO(10)$ models, and local F-theory $SO(10)$ models. To achieve gauge coupling unification, we introduce one pair of vector-like fermions, which form complete $SU(5)\times U(1)_X$ representation. Proton lifetime is around $5\times 10^{35}$ years, neutrino masses and mixing can be explained via seesaw mechanism, baryon asymmetry can be generated via leptogenesis, and vacuum stability problem can be solved as well. In particular, we propose that inflaton and dark matter particle can be unified to a real scalar field with $Z_2$ symmetry, which is not an axion and does not have the non-minimal coupling to gravity. Such kind of scenarios can be applied to the generic scalar dark matter models. Also, we find that the vector-like particle corrections to the $B_s^0$ masses can be about 6.6%, while their corrections to the $K^0$ and $B_d^0$ masses are negligible.Comment: 5 pages, 4 figures;V2: published versio