<div>In the first part of the present work, we will study the harmonic maps onto Teichm\"uller space. We will formulate a general Bochner type formula for harmonic maps into Teichm\"uller space. We will also prove the existence theorem of equivariant harmonic maps from a symmetric space with finite volume into its Weil-Petersson completion $\overline{\mathcal{T}}$, by deforming an almost finite energy map in the sense of Saper into a finite energy map.</div><div><br></div><div>In the second part of the work, we discuss the superrigidity of mapping class group. We will provide a geometric proof of both the high rank and the rank one superrigidity of mapping class groups due to Farb-Masur and Yeung. </div>

In this dissertation we considered a nonlinear sigma model with gravitino field. This is a supersymmetric extension of the nonlinear sigma model in the string theory, and we set up the geometric model using commuting variables, such that we could analyze it using the tools from calculus of variations. We introduced an action functional which corresponds to the super harmonic map functional, which has four arguments: a map between Riemannian manifolds, a vector spinor, a Riemannian metric and a gravitino. After getting the total variation formula, we considered the symmetries that the action functional possesses. By Noether's principle these families of symmetries induces conservation laws, which help to interpret the energy-momentum tensor and the supercurrent as holomorphic sections of some complex bundle. We also discussed the supersymmetry of our model. It turns out that the supersymmetry only remains in some particular cases, which is still useful in the analysis. Then we defined the weak solution in the distributional sense, and using Riesz potential estimates and Riviere regularity theory, we could improve the regularity of the weak solutions. More precisely, when the Riemannian metric and the gravitino are smooth, then any weak solution is actually smooth; and when the gravitino are coarse but subcritical, we can still show that the weak solutions are Holder continuous. Next we considered the compactness of solutions with bounded energies. We showed the small energy regularity on local domains and gap properties on the global surface. We also established the Pohozaev identities and thus showed the removable singularity theorem. Finally, for a sequence of solutions of uniformly bounded energies with respect to a converging sequence of gravitino fields, we showed that they converges weakly. Actually away from finite points, the convergence is strong and at those points, the energies concentrate. After a rescaling, each of these points corresponds to finitely some Dirac-harmonic maps with curvature terms defined on the Riemann sphere. Moreover, we established the energy identities along the weakly convergent sequences modulo these bubbles.

We develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. We apply our method to the special case where the harmonic map is the Gauss map of a minimal submanifold and the complete manifold is a Grassmannian. With the help of a result by Allard [Allard, W. K. (1972). On the first variation of a varifold. Annals of mathematics, 417-491.], we can study the graph case and have an approach to prove Bernstein-type theorems. This enables us to extend Moser’s Bernstein theorem [Moser, J. (1961). On Harnack's theorem for elliptic differential equations. Communications on Pure and Applied Mathematics, 14(3), 577-591.] to codimension two, i.e., a minimal p-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a p-plane.

This dissertation considers several problems related to Ricci flow, including the existence and behavior of solutions. The first goal is to obtain explicit, coordinate-based descriptions of Ricci flow solutions--especially those corresponding to Ricci solitons--on two classes of nilpotent Lie groups. On the odd-dimensional classical Heisenberg groups, we determine the asymptotics of Ricci flow starting at any metric, and use Lott's blowdown method to demonstrate convergence to soliton metrics. On the groups of real unitriangular matrices, which are more complicated, we describe the solitons and corresponding solutions using a suitable ansatz. Next, we consider solsolitons involving the nilsolitons in the Heisenberg case above. This uses work of Lauret, which characterizes solsolitons as certain extensions of nilsolitons, and work of Will, which demonstrates that the space of solsolitons extensions of a given nilsoliton is parametrized by the quotient of a Grassmannian by a finite group. We determine these spaces of solsoliton extensions of Heisenberg nilsolitons, and we also explicitly describe many-parameter families of these solsolitons in dimensions greater than three. Finally, we explore Ricci flow coupled with harmonic map flow, both as it arises naturally in certain bundle constructions related to Ricci flow and as a geometric flow in its own right. In the first case, we generalize a theorem of Knopf that demonstrates convergence and stability of certain locally R[superscript N]-invariant Ricci flow solutions. In the second case, we prove a version of Hamilton's compactness theorem for the coupled flow, and then generalize it to the category of etale Riemannian groupoids. We also provide a detailed example of solutions to the flow on the three-dimensional Heisenberg group. / text

We set out to analyze 5D stationary and bi-axisymmetric solutions to the vacuum Einstein equations. These are in the cohomogeneity 2 setting where the orbit space is a right half plane. They can have a wide range of behaviour at the boundary of the orbit space. The goal is to understand in detail the soliton example in Khuri, Weinstein and Yamada's paper ``5-dimensional space-periodic solutions of the static vacuum Einstein equations". This example is periodic and has alternating axis rods as its boundary data. We start by deriving the harmonic equations which determines the behaviour of the metric in the interior of the orbit space. Then we analyze what conditions the boundary data imposes on the metric. These are called the smoothness conditions which we derive for solely the alternating axis rod case. We show that with an ellipticity assumption they predict that the twist potentials are constant and that the metric is of the form which appears in Khuri, Weinstein and Yamada's paper. We then analyze the Schwarzschild metric in its standard form which is cohomogeneity 1 and its Weyl form which is cohomogeneity 2. This Weyl form can be made periodic and this serves as an inspiration for the examples in Khuri, Weinstein and Yamada's paper. Finally we analyze the soliton example in detail and show that it satisfies the smoothness conditions. We then provide a new example which has a single axis rod on the boundary with non-constant twist potentials but that is missing a point on the boundary. / Thesis / Master of Science (MSc) / We study the geometry of 5D blackholes. These blackholes are idealized by certain spatial symmetries and time invariance. They are solutions to the vacuum Einstein equations. The unique characteristic of these blackholes is the range of behaviour they may exhibit at the boundary of the domain of outer communication. There could be a standard event horizon called a horizon rod or an axis rod where a certain part of the spatial symmetry becomes trivial. In this thesis we start by deriving the harmonic map equations which are satisfied in the interior of the domain of communication. Then we show how this boundary data affects the metric through the smoothness conditions. We then analyze the soliton example in a paper by Khuri, Weinstein and Yamada and show that it respects the smoothness conditions. We then provide a new example which is interesting in the fact it has non-constant twist potentials.

Natural surfaces are usually associated with feature graphs, such as the cortical surface with anatomical atlas structure. Such a feature graph subdivides the whole surface into meaningful sub-regions. Existing brain mapping and registration methods did not integrate anatomical atlas structures. As a result, with existing brain mappings, it is difficult to visualize and compare the atlas structures. And also existing brain registration methods can not guarantee the best possible alignment of the cortical regions which can help computing more accurate shape similarity metrics for neurodegenerative disease analysis, e.g., Alzheimer’s disease (AD) classification. Also, not much attention has been paid to tackle surface parameterization and registration with graph constraints in a rigorous way which have many applications in graphics, e.g., surface and image morphing. This dissertation explores structural mappings for shape analysis of surfaces using the feature graphs as constraints. (1) First, we propose structural brain mapping which maps the brain cortical surface onto a planar convex domain using Tutte embedding of a novel atlas graph and harmonic map with atlas graph constraints to facilitate visualization and comparison between the atlas structures. (2) Next, we propose a novel brain registration technique based on an intrinsic atlas-constrained harmonic map which provides the best possible alignment of the cortical regions. (3) After that, the proposed brain registration technique has been applied to compute shape similarity metrics for AD classification. (4) Finally, we propose techniques to compute intrinsic graph-constrained parameterization and registration for general genus-0 surfaces which have been used in surface and image morphing applications.