Comparison theorems for manifolds with mean convex boundary

Let M-n be an n-dimensional Riemannian manifold with boundary partial derivative M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k is an element of R, we give a sharp estimate of the upper bound of rho(x) = d(x, partial derivative M), in terms of the mean curvature bound of the boundary. When partial derivative M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kahler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kahler manifold and also estimate the first eigenvalue of the real Laplacian.SCI(E)[email protected]

A simple finite-difference modification for improving accuracy near a corner in heat flow problems

Traditional numerical methods of solution of heat flow problems are subject to inaccuracies near sharp corners, where the derivatives of the exact solution may "become unbounded (Jeffreys1). 2Previous methods of overcoming this difficulty (Motz, Woods3

Effects of turbulent dust grain motion to interstellar chemistry

Theoretical studies have revealed that dust grains are usually moving fast through the turbulent interstellar gas, which could have significant effects upon interstellar chemistry by modifying grain accretion. This effect is investigated in this work on the basis of numerical gas-grain chemical modeling. Major features of the grain motion effect in the typical environment of dark clouds (DC) can be summarised as follows: 1) decrease of gas-phase (both neutral and ionic) abundances and increase of surface abundances by up to 2-3 orders of magnitude; 2) shifts of the existing chemical jumps to earlier evolution ages for gas-phase species and to later ages for surface species by factors of about ten; 3) a few exceptional cases in which some species turn out to be insensitive to this effect and some other species can show opposite behaviors too. These effects usually begin to emerge from a typical DC model age of about 10^5 yr. The grain motion in a typical cold neutral medium (CNM) can help overcome the Coulomb repulsive barrier to enable effective accretion of cations onto positively charged grains. As a result, the grain motion greatly enhances the abundances of some gas-phase and surface species by factors up to 2-6 or more orders of magnitude in the CNM model. The grain motion effect in a typical molecular cloud (MC) is intermediate between that of the DC and CNM models, but with weaker strength. The grain motion is found to be important to consider in chemical simulations of typical interstellar medium.Comment: 20 pages, 10 figures and 2 table

Chiral structures of lander molecules on Cu(100)

Supramolecular assemblies of lander molecules (C$_{90}$H$_{98}$) on Cu(100) are investigated with low-temperature scanning tunneling microscopy. The energetically most favourable conformation of the adsorbed molecule is found to exist in two mirror symmetric enantiomers or conformers. At low coverage, the molecules align in enantiomerically pure chains along the chiral directions $[01\bar{2}],[02\bar{1}],[012]$ and $[021]$. The arrangement is proposed to be mainly governed by intermolecular van-der-Waals interaction. At higher coverages, the molecular chains arrange into chiral domains, for which a structural model is presented.Comment: to appear in Nanotechnology vol. 15 (2004

Variational formulas of higher order mean curvatures

In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total $2p$-th mean curvature functional $\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold $N^{n+m}$ for $p=0,1,...,[\frac{n}{2}]$. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional $\mathcal {M}_{2p}$, called relatively $2p$-minimal submanifolds, for all $p$. At last, we discuss the relations between relatively $2p$-minimal submanifolds and austere submanifolds in real space forms, as well as a special variational problem.Comment: 13 pages, to appear in SCIENCE CHINA Mathematics 201