Wong, Yat Sen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 71-72). / Abstract also in Chinese. / Chapter 1 --- Lorentz Manifolds --- p.6 / Chapter 1.1 --- Preliminaries --- p.6 / Chapter 1.2 --- "Parallel Translation, geodesics and exponential map" --- p.9 / Chapter 1.3 --- "Curvature, frame field and Ricci curvature" --- p.14 / Chapter 1.4 --- Two-parameter maps --- p.15 / Chapter 2 --- Causal character in Lorentz geometry --- p.17 / Chapter 2.1 --- The Gauss Lemma --- p.17 / Chapter 2.2 --- Local causal character --- p.19 / Chapter 2.3 --- Timecones --- p.20 / Chapter 2.4 --- Local Lorentz Geometry --- p.21 / Chapter 3 --- Calculus of variations --- p.24 / Chapter 3.1 --- Jacobi fields --- p.24 / Chapter 3.2 --- Lorentz submanifolds --- p.25 / Chapter 3.3 --- The endmanifold case --- p.26 / Chapter 3.4 --- Focal points --- p.27 / Chapter 3.5 --- A causality theorem --- p.28 / Chapter 4 --- Causality in Lorentz manifolds --- p.32 / Chapter 4.1 --- Causality relations --- p.32 / Chapter 4.2 --- Quasi-limits --- p.35 / Chapter 4.3 --- Causality conditions --- p.37 / Chapter 4.4 --- Time separation --- p.39 / Chapter 4.5 --- Achronal sets --- p.42 / Chapter 4.6 --- Cauchy hypersurfaces --- p.43 / Chapter 4.7 --- Cauchy developments --- p.45 / Chapter 4.8 --- Spacelike hypersurfaces --- p.49 / Chapter 4.9 --- Cauchy horizons --- p.52 / Chapter 4.10 --- Hawking´ةs Singularity Theorem --- p.56 / Chapter 4.11 --- Penrose´ةs Singularity Theorem --- p.58 / Chapter 5 --- "Monge-Ampere equations, the Bergman kernel, and geometry of pesudoconvex domains" --- p.62 / Chapter 5.1 --- Monge-Ampere equations --- p.62 / Chapter 5.2 --- Differential geometry on the boundary --- p.63 / Chapter 5.3 --- Computations --- p.66 / Bibliography --- p.71

In this thesis, we explore the role of planarity in mean curvature flow in higher codimension and investigate its implications for singularity formation in a certain class of flows. In Chapter 1, we show that the blow-ups of compact 𝑛-dimensional solutions to mean curvature flow in ℝⁿ initially satisfying the pinching condition |𝐴|² < c |𝐻|² for a suitable constant c = c(𝑛) must be codimension one. We do this by establishing a new a priori estimate via a maximum principle argument. In Chapter 2, we consider ancient solutions to the mean curvature flow in ℝⁿ⁺¹ (𝑛 ≥ 3) that are weakly convex, uniformly two-convex, and satisfy derivative estimates |∇𝐴| ≤ 𝛾1 |𝐻|², |∇² 𝐴| \leq 𝛾2 |𝐻|³. We show that such solutions are noncollapsed. The proof is an adaptation of the foundational work of Huisken and Sinestrari on the flow of two-convex hypersurfaces. As an application, in arbitrary codimension, we classify the singularity models of compact 𝑛-dimensional (𝑛 ≥ 5) solutions to the mean curvature flow in ℝⁿ that satisfy the pinching condition |𝐴|² < c |𝐻|² for c = min {1/𝑛-2, 3(𝑛+1)/2𝑛(𝑛+2)}. Using recent work of Brendle and Choi, together with the estimate of Chapter 1, we conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton. Finally, in Chapters 3 and 4, we prove a canonical neighborhood theorem for the mean curvature flow of compact 𝑛-dimensional submanifolds in ℝⁿ (𝑛 ≥ 5) satisfying a pinching condition |𝐴|² < c |𝐻|² for $c = min {1/𝑛-2, 3(𝑛+1)/2𝑛(𝑛+2)}. We first discuss, in some detail, a well-known compactness theorem of the mean curvature flow. Then, adapting an argument of Perelman and using the conclusions of Chapter 2, we characterize regions of high curvature in the pinched solutions of the mean curvature flow under consideration.

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The goal of this paper is to establish necessary and sufficient conditions for a subgroup of the full homeomorphism group of a manifold to be the stability group of a point in the underlying space. Such subgroups are useful in identifying the underlying space in terms of its homeomorphism group even in cases in which this space is not necessarily a manifold. Thus, stability groups are useful in classifying various spaces.

The representation theory of infinite dimensional groups is in its infancy. This paper is an attempt to apply the orbit method to a particular infinite dimensional group, the spin extension of the restricted unitary group. Our main contribution is in showing that various homogeneous spaces for this group admit measures which can be used to realize the unitary structure for the standard modules.

The results from this dissertation are a computation of ultrapowers by supercompactness measures and concepts related to such measures. The second chapter gives an overview of the basic ideas required to carry out the computations. Included are preliminary ideas connected to measures, and the supercompactness measures. Order type results are also considered in this chapter. In chapter III we give an alternate characterization of 2 using the notion of iterated ordinal measures. Basic facts related to this characterization are also considered here. The remaining chapters are devoted to finding bounds fwith arguments taking place both inside and outside the ultrapowers. Conditions related to the upper bound are given in chapter VI.

Let $S$ be a $2n$-dimensional complex manifold equipped with a line bundle with a real-analytic complex connection such that its curvature is of type $(1,1)$, and with a real analytic h-projective structure such that its h-projective curvature is of type $(1,1)$. For $n=1$ we assume that $S$ is equipped with a real-analytic M\"obius structure. Using the structure on $S$, we construct a twistor space of a quaternionic $4n$-manifold $M$. We show that $M$ can be identified locally with a neighbourhood of the zero section of the twisted (by a unitary line bundle) tangent bundle of $S$ and that $M$ admits a quaternionic $S^1$ action given by unit scalar multiplication in the fibres. We show that $S$ is a totally complex submanifold of $M$ and that a choice of a connection $D$ in the h-projective class on $S$ gives extensions of a complex structure from $S$ to $M$. For any such extension, using $D$, we construct a hyperplane distribution on $Z$ which corresponds to the unique quaternionic connection on $M$ preserving the extended complex structure. We show that, in a special case, the construction gives the Feix--Kaledin construction of hypercomplex manifolds, which includes the construction of hyperk\"ahler metrics on cotangent bundles. We also give an example in which the construction gives the quaternion-K\"ahler manifold $\mathbb{HP}^n$ which is not hyperk\"ahler. We show that the same construction and results can be obtained for $n=1$. By convention, in this case, $M$ is a self-dual conformal $4$-manifold and from Jones--Tod correspondence we know that the quotient $B$ of $M$ by an $S^1$ action is an asymptotically hyperbolic Einstein--Weyl manifold. Using a result of LeBrun \cite{Le}, we prove that $B$ is an asymptotically hyperbolic Einstein--Weyl manifold. We also give a natural construction of a minitwistor space $T$ of an asymptotically hyperbolic Einstein--Weyl manifold directly from $S$, such that $T$ is the Jones--Tod quotient of $Z$. As a consequence, we deduce that the Einstein--Weyl manifold constructed using $T$ is equipped with a distinguished Gauduchon gauge.

Lau Chi Hin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 88-91). / Chapter 1 --- Introduction --- p.4 / Chapter 1.1 --- Rough Isometries --- p.4 / Chapter 1.2 --- Discrete approximation of Riemannian manifolds --- p.8 / Chapter 2 --- Basic Properties of Rough Isometries --- p.19 / Chapter 2.1 --- Volume growth rate --- p.19 / Chapter 2.2 --- Sobolev Inequalities --- p.25 / Chapter 2.3 --- Poincare Inequality --- p.32 / Chapter 3 --- Parabolic Harnack Inequality --- p.39 / Chapter 3.1 --- Parabolic Harnack Inequality --- p.39 / Chapter 4 --- Parabolicity and Liouville Dp-property --- p.58 / Chapter 4.1 --- Parabolicity --- p.58 / Chapter 4.2 --- Liouville Dp-property --- p.67