A Gauge field Induced by the Global Gauge Invariance of Action Integral

As a general rule, it is considered that the global gauge invariance of an action integral does not cause the occurrence of gauge field. However, in this paper we demonstrate that when the so-called localized assumption is excluded, the gauge field will be induced by the global gauge invariance of the action integral. An example is given to support this conclusion.Comment: 13 pages. Some typing errors are corrected and the format is update

Erratum to: An Entropy Functional for Riemann-Cartan Space-Times

We correct the entropy functional constructed in Int. J. Theor. Phys. 51:362 (2012). The 'on-shell' functional one obtains from this correct functional possesses a holographic structure without imposing any constraint on the spin-angular momentum tensor of matter, in contrast to the conclusion made in the above paper.Comment: 15 pages. These are the preprints of the original paper and its erratum published in Int. J. Theor. Phy

Mappings of least Dirichlet energy and their Hopf differentials

The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class $\bar{\mathscr H}_2(X, Y)$ of strong limits of homeomorphisms in the Sobolev space $W^{1,2}(X, Y)$, a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential $h_z \bar{h_{\bar{z}}} dz \otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h \in \bar{\mathscr H}_2(X, Y)$ is energy-minimal if and only if its Hopf-differential is analytic in $X$ and real along the boundary of $X$. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in $X$. Nevertheless, cracks are triggered only by the points in the boundary of $Y$ where $Y$ fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of $X$ toward the interior of $X$ where they eventually terminate before making a crosscut.Comment: 51 pages, 4 figure