Research on the Regularity of Solutions for Geometric Variational Problems

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The variational problems for the energy functional defined for maps between Riemmannian manifolds have been studied by many mathematicians.Namely, about harmonic maps between Riemannian manifolds, we have many deep results.Recently, some generalisations of harmonic maps attract the interest of several reseachers.In this research, we considered some generalised notion of harmonic maps and got some results on their regularityIn the year 2000, we treated harmonic maps with potentials, and got their existence and regularity results.In the year 2001, we considered more generalised notion of harmonic maps so-called F-farmonic tnays and get their existence and partial regularity resultsIn the year 2002, we treated harmonic maps into Finsler manidolds.Finsler manifolds are natural generalization of Riemannian ones.P.Centore defined the energy of a map between Finsler manifolds.We used Centores definition and specialized it for the case that the source manifold is Eucldean space. We calculated the Euler-Lagrange equation of it and got the equation for hannonic maps from an. Euclidean space R^m to a Finsler manifold. Moreover, we got a partial regularity result for energy minimizing map from R^m to a Finsler manifold for the case that m=3,4.More precisely, we proved that for the above cases the enegy minimizing maps are Wilder continuous outside the singular set whose Hausdorff dimension is less that m-2.