The study of bifurcations of vectorfields is concerned with changes in qualitative behaviour that can occur when a non-structurally stable vectorfield is perturbed. In a sense, this is the study of how such a vectorfield fails to be structurally stable. Finding a systematic approach to the study of such questions is a difficult problem. One approach to bifurcations of vectorfields, known as "generic bifurcation theory," is the subject of much of the work of Sotomayor (Sotomayor [1973a], Sotomayor [1973b],Sotomayor [1974]). This approach attempts to construct generic families of k-parameter vectorfields (usually for k=1), for which all the bifurcations can be described. In Sotomayor [1973a] it is stated that the vectorfields associated with the "generic" bifurcations of individual critical elements for k-parameter vectorfields form submanifolds of codimension ≤ k of the Banach space ϰʳ (M) of vectorfields on a compact manifold M. The bifurcations associated with one of these submanifolds of codimension-k are called generic codimension-k bifurcations. In Sotomayor [1974] the construction of these submanifolds and the description of the associated bifurcations of codimension-1 for vectorfields on two dimensional manifolds is presented in detail. The bifurcations that occur are due to the parameterised vectorfield crossing one of these manifolds transversely as the parameter changes. Abraham and Robbin used transversality results for evaluation maps to prove the Kupka-Smale theorem in Abraham and Robbin [1967]. In this thesis, we shall show how an extension of these evaluation transversality techniques will allow us to construct the submanifolds of ϰʳ (M) associated with one type of generic bifurcation of critical elements, and we shall consider how this approach might be extended to include the other well known generic bifurcations. For saddle-node type bifurcations of critical points, we will show that the changes in qualitative behaviour are related to geometric properties of the submanifold Σ₀ of ϰʳ (M) x M, where Σ₀ is the pull-back of the set of zero vectors-or zero section-by the evaluation map for vectorfields. We will look at the relationship between the Taylor series of a vector-field X at a critical point ⍴ and the geometry of Σ₀ at the corresponding point (X,⍴) of ϰʳ (M) x M. This will motivate the non-degeneracy conditions for the saddle-node bifurcations, and will provide a more general geometric picture of this approach to studying bifurcations of critical points. Finally, we shall consider how this approach might be generalised to include other bifurcations of critical elements. / Science, Faculty of / Mathematics, Department of / Graduate

In this thesis, we study three problems in complex geometry. In the first part, we study the behavior of the Kahler-Ricci flow on complete non-compact manifolds with negative holomorphic curvature. We show that Kahler-Ricci flow converges to a Kahler-Einstein metric when the initial manifold admits a suitable exhaustion function, thus improving upon a result of D. Wu and S.T. Yau. These results are partly obtained in joint work with S. Huang, M.-C. Lee and L.-F. Tam. In the second part of this thesis, we introduce a new Kodaira-Bochner type formula for closed (1, 1)-form in non-Kahler geometry. Based on this new formula, We propose a new curvature positivity condition in non-Kahler manifolds and proved a strong rigidity type theorem for manifolds satisfying this curvature positivity condition. We also find interesting examples non-Kahler manifolds satisfying the curvature positivity condition in a class of manifolds called Vaisman manifolds. In the third part of this thesis, we study the degenerations of asymptotically conical Calabi-Yau manifolds as the Kahler class degenerates to a non-Kahler class. Under suitable hypothesis, we prove the convergence of asymptotically conical Calabi-Yau metrics to a singular asymptotically conical Calabi-Yau current with compactly supported singularities. Using this, we construct singular asymptotically conical Calabi-Yau metrics on non-compact singular varieties and identify the topology of these singular metrics with the singular variety. We also give some interpretations of these asymptotically conical Calabi-Yau metrics from the point of view of physics. These results are obtained in joint work with T. Collins and B. Guo.

In this thesis, we explore short time existence and uniqueness of solutions to the Ricci flow on manifolds with boundary, as well as the preservation of natural curvature positivity conditions along the flow. In chapter 2, we establish the existence and uniqueness for linear parabolic systems on vector bundles for Hölder continuous initial data. We introduce appropriate weighted parabolic Hölder spaces to study the existence and uniqueness problem. Having developed the linear theory, we apply it to establish the existence and uniqueness for the Ricci-DeTurck flow, the harmonic map heat flow, and the Ricci flow with Hölder continuous initial data in Chapter 3. In chapter 4, we discuss a general preservation result concerning the preservation of various curvature conditions during boundary deformation. Using a perturbation argument, we construct a family of metrics which interpolate between two metrics that agree on the boundary, and such family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. The results from chapters 2 through 4 will be utilized in proving the Main Theorems in chapter 5. In particular, we construct canonical solutions to the Ricci flow on manifolds with boundary from canonical solutions to the Ricci flow on closed manifolds with Hölder continuous initial data via doubling.

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This work was motivated by the well-known question: "Does there exist a nondesarguesian projective plane of prime order?" For a prime p < 11, there is only the pappian plane of order p. Hence, such planes are indeed desarguesian. Thus, it is of interest to examine whether there are non-desarguesian planes of order 11. A suggestion by Ascher Wagner in 1985 was made to Spyros S. Magliveras: "Begin with a non-desarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations, and check whether one of these is non-desarguesian." In this manuscript we use a group-theoretic methodology to determine the subplane structures of some non-desarguesian planes. In particular, we determine orbit representatives of all proper Q-subplanes both of a Veblen-Wedderburn (VW) plane of order 121 and of the Hughes plane of order 121, under their full collineation groups. In PI, there are 13 orbits of Baer subplanes, all of which are desarguesian, and approximately 3000 orbits of Fano subplanes. In Sigma , there are 8 orbits of Baer subplanes, all of which are desarguesian, 2 orbits of subplanes of order 3, and at most 408; 075 distinct Fano subplanes. In addition to the above results, we also study the subplane structures of some non-desarguesian planes, such as the Hall plane of order 25, the Hughes planes of order 25 and 49, and the Figueora planes of order 27 and 125. A surprising discovery by L. Puccio and M. J. de Resmini was the existence of a plane of order 3 in the Hughes plane of order 25. We generalize this result, showing that there are subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q 5 (mod 6). Furthermore, we analyze the structure of the full collineation groups of certain Veblen- Wedderburn (VW) planes of orders 25, 49 and 121, and discuss how to recover the planes from their collineation groups. / by Cafer Caliskan. / Thesis (Ph.D.)--Florida Atlantic University, 2010. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web.

by Fan Xuqian. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 99-100). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- The Theorem of Efimov --- p.7 / Chapter 2.1 --- The Idea of the Proof of the Efimov's Theorem --- p.8 / Chapter 2.2 --- Proof of the Efimov's Main Lemma --- p.12 / Chapter 2.3 --- Proof of Lemma 2.3 --- p.48 / Chapter 2.4 --- Proof of Lemma 2.4 --- p.52 / Chapter 3 --- Isometric Immersion into R3 of Complete Surfaces with Negative Curvature --- p.62 / Chapter 3.1 --- The Sketch of the Proof of Theorem 3.1 --- p.66 / Chapter 3.2 --- Proof of Lemma 3.4 --- p.75 / Chapter 3.3 --- Proof of Lemma 3.5 --- p.76 / Chapter 3.4 --- Proof of Lemma 3.6 --- p.86 / Chapter 3.5 --- Proof of Lemma 3.7 --- p.89 / Chapter 3.6 --- The Geometric Properties of the Immersion --- p.95

by He Zhaokui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 81-82). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.8 / Chapter 2.1 --- The Laplacian of a compact manifold --- p.8 / Chapter 2.2 --- The Laplacian of a graph --- p.9 / Chapter 2.3 --- Some basic facts about the eigenvalues of a graph --- p.13 / Chapter 3 --- Bound of the first non-zero eigenvalue in terms of Cheeger constant --- p.18 / Chapter 3.1 --- The Cheeger constant --- p.18 / Chapter 3.2 --- The Cheeger inequality of a compact manifold --- p.19 / Chapter 3.3 --- The Cheeger inequality of a graph --- p.23 / Chapter 4 --- Diameters and eigenvalues --- p.27 / Chapter 4.1 --- Some facts --- p.27 / Chapter 4.2 --- Estimate the eigenvalues of graphs --- p.29 / Chapter 4.3 --- The heat kernel of compact manifolds --- p.34 / Chapter 4.4 --- Estimate the eigenvalues of manifolds --- p.35 / Chapter 5 --- Harnack inequality and eigenvalues on homogeneous graphs --- p.40 / Chapter 5.1 --- Preliminaries --- p.40 / Chapter 5.2 --- The Neumann eigenvalue of a subgraph --- p.41 / Chapter 5.3 --- The Harnack inequality --- p.44 / Chapter 5.4 --- A lower bound of the first non-zero eigenvalue --- p.52 / Chapter 6 --- Harnack inequality and eigenvalues on compact man- ifolds --- p.54 / Chapter 6.1 --- Gradient estimate --- p.54 / Chapter 6.2 --- Lower bounds for the first non-zero eigenvalue --- p.59 / Chapter 7 --- Heat kernel and eigenvalues of graphs --- p.63 / Chapter 7.1 --- The heat kernel of a graph --- p.54 / Chapter 7.2 --- Lower bounds for eigenvalues --- p.70 / Chapter 8 --- Estimate the eigenvalues of a compact manifold --- p.73 / Chapter 8.1 --- An isoperimetric constant --- p.75 / Chapter 8.2 --- A lower estimate for the (m + l)-st eigenvalue --- p.77

by Yiu Chun Chit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 81-82). / Chapter 1 --- Introduction --- p.7 / Chapter 1.1 --- Riemannian Manifolds --- p.7 / Chapter 1.1.1 --- Completeness --- p.8 / Chapter 1.1.2 --- Curvature tensor --- p.9 / Chapter 1.1.3 --- Holonomy --- p.11 / Chapter 1.2 --- Simply-connected Manifold of Nonpositive Sectional Curvature --- p.11 / Chapter 1.2.1 --- Topological structure --- p.12 / Chapter 1.2.2 --- Basic geometric properties --- p.13 / Chapter 1.2.3 --- Examples of nonpositively curved manifold --- p.20 / Chapter 1.2.4 --- Convexity properties --- p.23 / Chapter 1.2.5 --- Points at infinity for M --- p.27 / Chapter 2 --- Symmetric Spaces --- p.36 / Chapter 2.1 --- Symmetric Spaces of Noncompact Type --- p.36 / Chapter 2.1.1 --- Symmetric diffeomorphisms --- p.36 / Chapter 2.1.2 --- Transvections in I(M) --- p.38 / Chapter 2.1.3 --- Symmetric spaces as coset manifolds G/K --- p.39 / Chapter 2.1.4 --- Metric on TpM and the adjoint representation of Lie group --- p.41 / Chapter 2.1.5 --- Curvature tensor of M --- p.43 / Chapter 2.1.6 --- Killing form and classification of symmetric spaces --- p.44 / Chapter 2.1.7 --- Holonomy of M at p --- p.44 / Chapter 2.1.8 --- Rank of a symmetric space M --- p.45 / Chapter 2.1.9 --- Regular and singular points at infinity --- p.46 / Chapter 2.2 --- "The Symmetric Space Mn = SL(n,R)/SO(n,R)" --- p.46 / Chapter 2.2.1 --- Metric on TIMn --- p.47 / Chapter 2.2.2 --- Geodesic and symmetries of Mn --- p.48 / Chapter 2.2.3 --- Curvature of Mn --- p.48 / Chapter 2.2.4 --- Rank and flats in Mn --- p.49 / Chapter 2.2.5 --- Holonomy of Mn at I --- p.49 / Chapter 2.2.6 --- Eigenvalue-flag pair for a point in Mn(∞ ) --- p.50 / Chapter 2.2.7 --- Action of I0(Mn) on Mn(∞ ) --- p.52 / Chapter 2.2.8 --- Flags in opposition --- p.53 / Chapter 2.2.9 --- Joining points at infinity --- p.53 / Chapter 3 --- Group Action --- p.56 / Chapter 3.1 --- Action of Isometries on M(oo) --- p.56 / Chapter 3.1.1 --- Fundamental group as a group of isometries --- p.56 / Chapter 3.1.2 --- Lattices --- p.58 / Chapter 3.1.3 --- Duality condition --- p.59 / Chapter 3.1.4 --- Geodesic flows --- p.61 / Chapter 3.2 --- Action of Geodesic Symmetries on M(oo) --- p.62 / Chapter 3.3 --- Rank --- p.66 / Chapter 3.3.1 --- Rank of a manifold of nonpositive curvature --- p.66 / Chapter 3.3.2 --- Rank of the fundamental group --- p.68 / Chapter 3.4 --- Rigidity Theorems of Locally Symmetric Spaces --- p.69

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

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