Hardt, Robert M.2009-06-042009-06-041995Li, Ming. "Harmonic maps, heat flows, currents and singular spaces." (1995) Diss., Rice University. <a href="https://hdl.handle.net/1911/16848">https://hdl.handle.net/1911/16848</a>.https://hdl.handle.net/1911/16848This thesis studies some problems in geometry and analysis with techniques developed from non-linear partial differential equations, variational calculus, geometric measure theory and topology. It consists of three independent parts: Chapter I. We study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at most n-2, where n is the dimension of the domain. We will also give an example of an energy minimizing map from a surface to a surface that has a singular point. Thus the n-2 dimension estimate is optimal, in contrast to the n-3 dimension estimate of Schoen-Uhlenbeck (SU) for compact targets. Chapter II. Here we study a new intersection homology theory for currents on a space X with cone-like singularities. This homology is given by a new mass functional $M\sb{p}$ associated with the perversity index p. For X, it pairs with the intersection homology of Goresky-MacPherson, as well as the $L\sp2$-cohomology of J. Cheeger. We also give a deformation theorem and then prove the existence of $M\sb{p}$-minimizing currents in a given intersection homology class. Chapter III. We construct a weak solution for the heat flow associated with various quasiconvex functionals into homogeneous spaces, in particular, the p-harmonic map heat flow for any $p > 1.$ Our proof generalizes previous works (CHN), (CH2) which treated the case for $p \ge 2$ where the target is a sphere.69 p.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.MathematicsThesisTHESIS MATH. 1995 LI