Topological and differentiable rigidity of submanifolds in space forms

Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean curvature of $M$, then $M$ is homeomorphic to a sphere. We also show that the pinching condition above is sharp. Moreover, we obtain a new differentiable sphere theorem for submanifolds with positive Ricci curvature.Comment: 12 page

The sphere theorems for manifolds with positive scalar curvature

Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>\sigma_{n}K_{\max}$, where $\sigma_n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if $M^n(n\geq3)$ is a compact manifold whose $(n-2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition $Ric^{(n-2)}_{\min}>\tau_n(n-2)R_0,$ where $\tau_n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker and the authors.Comment: 35 page

Efficient Volumetric Method of Moments for Modeling Plasmonic Thin-Film Solar Cells with Periodic Structures

Metallic nanoparticles (NPs) support localized surface plasmon resonances (LSPRs), which enable to concentrate sunlight at the active layer of solar cells. However, full-wave modeling of the plasmonic solar cells faces great challenges in terms of huge computational workload and bad matrix condition. It is tremendously difficult to accurately and efficiently simulate near-field multiple scattering effects from plasmonic NPs embedded into solar cells. In this work, a preconditioned volume integral equation (VIE) is proposed to model plasmonic organic solar cells (OSCs). The diagonal block preconditioner is applied to different material domains of the device structure. As a result, better convergence and higher computing efficiency are achieved. Moreover, the calculation is further accelerated by two-dimensional periodic Green's functions. Using the proposed method, the dependences of optical absorption on the wavelengths and incident angles are investigated. Angular responses of the plasmonic OSCs show the super-Lambertian absorption on the plasmon resonance but near-Lambertian absorption off the plasmon resonance. The volumetric method of moments and explored physical understanding are of great help to investigate the optical responses of OSCs.Comment: 11 pages, 6 figure

Hypertrophic Cardiomyopathy Complicated by Left Ventricular Apical Necrosis and Aneurysm in a Young Man: FDG-PET Findings

A 29-year old male was transferred to our hospital with an abnormal chest X-ray finding diagnosed as hypertrophic cardiomyopathy with apical necrosis and aneurysm formation. Four years after the initial hospitalization, we confirmed the aneurysm and necrosis using both integrated positron emission tomography (PET) and computed tomography (CT) scanning. The F-18 2-fluoro-2-deoxy-D-glucose (FDG) PET/CT enabled precise localization of the aneurysm, which was found to be composed of semi-lunar calcification of non-metabolic myocardium. A contrast-enhanced CT angiography showed an hour-glass appearance of the left ventricular cavity. The integrated PET/CT fusion scanner is a novel multimodality technology that allows for a comprehensive analysis of the anatomical and functional status of complex heart disease. Based on these findings, long standing mechanical and physiologic abnormalities may have led to chronic ischemia in the hypertrophied myocardium, induced necrosis and calcification at the cardiac apex

What is the Shape Effect on the (Hyper)polarizabilities? A Comparison Study on the Moebius, Cyclic, and Linear Nitrogen-Substituted Polyacenes

How does the framework shape influence the static polarizability (alpha_0) and the first hyperpolarizability (beta_0)? This work, for the first time, presents a comparison study at the MP2/6-31+G(d) level, by using the nitrogen-substituted polyacenes as models: the Moebius strip with a knot, the cyclic strip without knot, and the linear strip. Opening the knot of the Moebius strip to form the cyclic strip, it leads to the increase of the alpha_0 from 268 au to 323 au and the beta_0 value increases about three times from 393 (Moebius) to 1049 au (cyclic). Further, opening the cyclic strip to form the linear strip, the alpha_0 value increases from 323 au to 476 au. While the beta_0 value as well increases about three times from 1049 (cyclic) to 2814 au (linear). The changes in the static (hyper)polarizabilities are well explained by the geometrical differences among the Moebius, cyclic, and linear nitrogen-substituted polyacenes.Comment: 10 pages, 3 figure