Thesis (Ph. D.)--University of Wisconsin--Madison, 1971. / Vita. Typescript. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

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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令X 為個帶有凱勒形式(Kähler form ω) 以及全純四形式( holomorphic four- form Ω )的四維緊致卡拉比丘空間(Calabi-Yau manifolds) 。在一些假設條件下，通過研究Donaldson- Thomas方程所決定的模空間，我們定義了四維Donaldson-Thomas不變量。我們也對四維局部卡拉比丘空間(local Calabi-Yau four-folds) 定義了四維Donaldson-Thomas 不變量，並且將之聯繫到三維Fano空間的Donaldson- Thomas 不變量。在一些情況下，我們還研究了DT/GW不變量對應。最后，我們在模空間光滑時計算了一些四維Donaldson- Thomas不變量。 / Let X be a complex four-dimensional compact Calabi-Yau manifold equipped with a Kahler form ω and a holomorphic four-form Ω. Under certain assumptions, we de ne Donaldson-Thomas type deformation invariants by studying the moduli space of the solutions of Donaldson-Thomas equations on the given Calabi-Yau manifold. We also study sheaves counting on local Calabi-Yau four-folds. We relate the sheaves countings over X = KY with the Donaldson- Thomas invariants for the associated compact three-fold Y . In some specialcases, we prove the DT/GW correspondence for X. Finally, we compute the Donaldson-Thomas invariants of certain Calabi-Yau four-folds when the moduli spaces are smooth. / Detailed summary in vernacular field only. / Cao, Yalong. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 100-105). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- The *4 operator --- p.18 / Chapter 2.1 --- The *4 operator for bundles --- p.18 / Chapter 2.2 --- The *4 operator for general coherent sheaves --- p.20 / Chapter 3 --- Local Kuranishi structure of DT₄ moduli spaces --- p.22 / Chapter 4 --- Compactification of DT₄ moduli spaces --- p.34 / Chapter 4.1 --- Stable bundles compactification of DT₄ moduli spaces --- p.34 / Chapter 4.2 --- Attempted general compactification of DT₄ moduli spaces --- p.36 / Chapter 5 --- Virtual cycle construction --- p.39 / Chapter 5.1 --- Virtual cycle construction for DT₄ moduli spaces --- p.40 / Chapter 5.2 --- Virtual cycle construction for generalized DT₄ moduli spaces --- p.48 / Chapter 6 --- DT4 invariants for compactly supported sheaves on local CY₄ --- p.52 / Chapter 6.1 --- The case of X = KY --- p.52 / Chapter 6.2 --- The case of X = T*S --- p.57 / Chapter 7 --- DT₄ invariants on toric CY₄ via localization --- p.66 / Chapter 8 --- Computational examples --- p.70 / Chapter 8.1 --- DT₄=GW correspondence in some special cases --- p.71 / Chapter 8.1.1 --- The case of Hol(X) = SU(4) --- p.72 / Chapter 8.1.2 --- The case of Hol(X) = Sp(2) --- p.77 / Chapter 8.2 --- Some remarks on cosection localizations for hyper-kähler four-folds --- p.79 / Chapter 8.3 --- Li-Qin's examples --- p.80 / Chapter 8.4 --- Moduli space of ideal sheaves of one point --- p.83 / Chapter 9 --- Appendix --- p.85 / Chapter 9.1 --- Local Kuranishi models of Mc° --- p.85 / Chapter 9.2 --- Some remarks on the orientability of the determinant line bundles on the (generalized) DT₄ moduli spaces --- p.87 / Chapter 9.3 --- Seidel-Thomas twists --- p.90 / Chapter 9.4 --- A quiver representation of Mc --- p.92

Zhang Yi. / "May 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 141-146). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

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A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kähler. A similar result is proved for 8-manifolds. HyperKähler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kähler manifold (indefinite if the scalar curvature is negative). This construction is compatible with the quaternionic Kähler and hyperKähier quotient constructions and allows quaternionic Kähler geometry to be subsumed into the theory of hyperKähler manifolds. It is shown that the hyperKähler metrics that arise admit a certain type of SU(2)- action, possess functions which are Kähler potentials for each of the complex structures simultaneously and determine quaternionic Kähler structures via a variant of the moment map construction. Quaternionic Kähler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKähler metrics with circle actions on complex line bundles over Kähler-Einstein (complex) contact manifolds. Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKähler metrics defined by Kronheimer, are shown to give rise to quaternionic Kähler metrics and various examples of these metrics are identified. It is shown that any quaternionic Kähler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds. The twistor space structure of the projectivised nilpotent orbits is studied.

Bibliography: p. 135-138. / 140 p. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1998?