The main goal of this paper is to use the theory of exterior differential forms in deriving variations of the deformed Willmore energy in space forms and study the minimizers of the deformed Willmore energy in space forms. We derive both first and second order variations of deformed Willmore energy in space forms explicitly using moving frame method. We prove that the second order variation of deformed Willmore energy depends on the intrinsic Laplace Beltrami operator, the sectional curvature and some special operators along with mean and Gauss curvatures of the surface embedded in space forms, while the first order variation depends on the extrinsic Laplace Beltrami operator.
This paper examine the unconditional and conditional relationships between beta, size, B/M, E/P and returns in the CSE from 1995 to 2006 by using (FM) (1973) cross-sectional regression test. There is no evidence of a positive risk premium on beta when the unconditional relationship between beta and return is considered. However, when the market is split on the market excess returns, there is a significant positive (negative) relationship between beta and returns in up (down) markets for individual and portfolio of stocks. Further, there is a significant negative size-return and positive B/M-return relations, are unconditional and also insignificant E/P-return relation in the CSE. Thus, we cannot reject the usability of beta in explaining stock returns in the CSE. Beta has a significant relationship with stock returns subject to the condition of the market. Further, size and B/M are significant in explaining stock returns in the CSE and it doesn’t seem that they are affected by the condition of the market. Since these findings are important, are could be useful for making investment decisions.