On the Cartan Curvatures of a Null Curve in Minkowski Spacetime

We reconstruct the Cartan frame of a null curve in Minkowski spacetime for an arbitrary parameter, and we characterize pseudo-spherical null curves and Bertrand null curves.Mathematics Subject Classification(2000). 53B30, 53A04.     

Dupin超曲面的主曲率的重数

作者证明了如下结果 :设M是空间形式的闭定向Dupin超曲面 ,其截面曲率为正 ,M至少有两个不同主曲率 .如果除最小 (或最大 )主曲率外 ,M的其余主曲率均为常数 ,则最小 (或最大 )主曲率的重数大于 2 .     

First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Let be an -dimensional compact hypersurface with constant scalar curvature , , in a unit sphere . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral of the mean curvature . In this paper, we first study the eigenvalue of the Jacobi operator of . We derive an optimal upper bound for the first eigenvalue of , and this bound is attained if and only if is a totally umbilical and non-totally geodesic hypersurface or is a Riemannian product , .

     

A microscopic convexity principle for spacetime convex solutions of fully nonlinear parabolic equations

We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations. Under certain general structure condition, we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations. At last, we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.     

Comparison theorems in Lorentzian geometry and applications to spacelike hypersurfaces

In this paper we prove Hessian and Laplacian comparison theorems for the Lorentzian distance function in a spacetime with sectional (or Ricci) curvature bounded by a certain function by means of a comparison criterion for Riccati equations. Using these results, under suitable conditions, we are able to obtain some estimates on the higher order mean curvatures of spacelike hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic operators.     

Biharmonic hypersurfaces in 4‐dimensional space forms

We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4‐dimensional space forms (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Observers in spacetime and nonlocality

Characteristics of observers in relativity theory are critically examined. For field measurements in Minkowski spacetime, the Bohr‐Rosenfeld principle implies that the connection between actual (i.e., noninertial) and inertial observers must be nonlocal. Nonlocal electrodynamics of non‐uniformly rotating observers is discussed and the consequences of this theory for the phenomenon of spin‐rotation coupling are briefly explored.