Let M be a Riemannian manifold with metric g, and let P be a principal G-bundle over M having connection one-form a. One can define a modified version of the Ricci flow on P by fixing the size of the fiber. These equations are called the Ricci Yang-Mills flow, due to their coupling of the Ricci flow and the Yang-Mills heat flow. In this thesis, we derive the Ricci Yang-Mills flow and show that solutions exist for a short time and are unique. We study obstructions to the long-time existence of the flow and prove a compactness theorem for pointed solutions. We represent the Ricci Yang-Mills flow as a gradient flow and derive monotonicity formulas that can be used to study breather and soliton solutions. Finally, we use maximal regularity theory and ideas of Simonett concerning the asymptotic behavior of abstract quasilinear parabolic partial differential equations to study the stability of the Ricci Yang-Mills flow in dimension 2 at Einstein Yang-Mills metrics. / text

In this dissertation, we present some analysis of Ricci flow on complete noncompact manifolds. The first half of the dissertation concerns the formation of Type-II singularity in Ricci flow on [mathematical equation]. For each [mathematical equation] , we construct complete solutions to Ricci flow on [mathematical equation] which encounter global singularities at a finite time T such that the singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate [mathematical equation]. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on [mathematical equation] whose blow-ups near the origin converge uniformly to the Bryant soliton. In the second half of the dissertation, we fully analyze the structure of the Lichnerowicz Laplacian of a Bergman metric g[subscript B] on a complex hyperbolic space [mathematical equation] and establish the linear stability of the curvature-normalized Ricci flow at such a geometry in complex dimension [mathematical equation]. We then apply the maximal regularity theory for quasilinear parabolic systems to prove a dynamical stability result of Bergman metric on the complete noncompact CH[superscript m] under the curvature-normalized Ricci flow in complex dimension [mathematical equation]. We also prove a similar dynamical stability result on a smooth closed quotient manifold of [mathematical symbols]. In order to apply the maximal regularity theory, we define suitably weighted little Hölder spaces on a complete noncompact manifold and establish their interpolation properties. / text

An intrinsic geometric flow is an evolution of a Riemannian metric by a two-tensor. An extrinsic geometric flow is an evolution of an immersion of a manifold into Euclidean space. An extrinsic flow induces an evolution of a metric because any immersed manifold inherits a Riemannian metric from Euclidean space. In this paper we discuss the inverse problem of specifying an evolution of a metric and then seeking an extrinsic geometric flow which induces the given metric evolution. We limit our discussion to the case of manifolds that are rotationally symmetric and embeddable with codimension one. In this case, we reduce an intrinsic geometric flow to a plane curve evolution. In the specific cases we study, we are able to further simplify the evolution to an evolution of a function of one variable. We provide soliton equations and give proofs that some soliton metrics exist.

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 74-76). Issued in print and online. Available via ProQuest Digital Dissertations.

Thesis (Ph. D.)--Duke University, 2007. / Includes bibliographical references.

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Dans cette thèse nous nous intéressons aux flots de Monge-Ampère complexes, à leurs généralisations et à leurs applications géométriques sur les variétés hermitiennes compactes. Dans les deux premiers chapitres, nous prouvons qu'un flot de Monge-Ampère complexe sur une variété hermitienne compacte peut être exécuté à partir d'une condition initiale arbitraire avec un nombre Lelong nul en tous points. En utilisant cette propriété, nous con- firmons une conjecture de Tosatti-Weinkove: le flot de Chern-Ricci effectue une contraction chirurgicale canonique. Enfin, nous étudions une généralisation du flot de Chern-Ricci sur des variétés hermitiennes compactes, le flot de Chern-Ricci tordu. Cette partie a donné lieu à deux publications indépendantes. Dans le troisième chapitre, une notion de C -sous-solution parabolique est introduite pour les équations paraboliques, étendant la théorie des C -sous-solutions développée récem- ment par B. Guan et plus spécifiquement G. Székelyhidi pour les équations elliptiques. La théorie parabolique qui en résulte fournit une approche unifiée et pratique pour l'étude de nombreux flots géométriques. Il s'agit ici d'une collaboration avec Duong H. Phong (Université Columbia ) Dans le quatrième chapitre, une approche de viscosité est introduite pour le problème de Dirichlet associé aux équations complexes de type hessienne sur les domaines de Cn. Les arguments sont modélisés sur la théorie des solutions de viscosité pour les équations réelles de type hessienne développées par Trudinger. En conséquence, nous résolvons le problème de Dirichlet pour les équations de quotient de hessiennes et lagrangiennes spéciales. Nous établissons également des résultats de régularité de base pour les solutions. Il s'agit ici d'une collaboration avec Sl-awomir Dinew (Université Jagellonne) et Hoang-Son Do (Institut de Mathématiques de Hanoi). / In this thesis we study the complex Monge-Ampère flows, and their generalizations and geometric applications on compact Hermitian manifods. In the first two chapters, we prove that a general complex Monge-Ampère flow on a compact Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti- Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow. This part gave rise to two independent publications. In the third chapter, a notion of parabolic C -subsolution is introduced for parabolic non-linear equations, extending the theory of C -subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows. This part is a joint work with Duong H. Phong (Columbia University) In the fourth chapter, a viscosity approach is introduced for the Dirichlet problem associated to complex Hessian type equations on domains in Cn. The arguments are modelled on the theory of viscosity solutions for real Hessian type equations developed by Trudinger. As consequence we solve the Dirichlet problem for the Hessian quotient and special Lagrangian equations. We also establish basic regularity results for the solutions. This part is a joint work with Sl-awomir Dinew (Jagiellonian University) and Hoang-Son Do (Hanoi Institute of Mathematics).

In this thesis, we study the Ricci flow and Ricci soliton equations on Riemannian manifolds which admit a certain degree of symmetry. More precisely, we investigate the Ricci soliton equation on connected Riemannian manifolds, which carry a cohomogeneity one action by a compact Lie group of isometries, and the Ricci flow equation for invariant metrics on a certain class of compact and connected homogeneous spaces. In the first case, we prove that the initial value problem for a cohomogeneity one gradient Ricci soliton around a singular orbit of the group action always has a solution, under a technical assumption. However, this solution is in general not unique. This is a generalisation of the analogous result for the Einstein equation, which was proved by Eschenburg and Wang in their paper "Initial value problem for cohomogeneity one Einstein metrics". In the second case, by studying the corresponding system of nonlinear ODEs, we identify a class of singular behaviours for the homogeneous Ricci flow on these spaces. The singular behaviours that we find all correspond to type I singularities, which are modelled on rigid shrinking solitons. In the case where the isotropy representation decomposes into two invariant irreducible inequivalent summands, we also investigate the existence of ancient solutions and relate this to the existence and non existence of invariant Einstein metrics. Furthermore, in this special case, we also allow the initial metric to be pseudo- Riemannian and we investigate the existence of immortal solutions. Finally, we study the behaviour of the scalar curvature for this more general situation and show that in the Riemannian case it always has to turn positive in finite time, if it was negative initially. By contrast, in the pseudo-Riemannian case, there are certain initial conditions which preserve negativity of the scalar curvature.

Cette these se compose de quatre chapîtres et une annexe. Le premier chapître est consacre à des idées fondamentales de la theorie du flot de Ricci, qui montre comment nos travaux sont reliés a l'histoire entière. Dans le deuxième chapître, nous construisons une solution du flot de Ricci sur une variete a symétrie de rotation de telle sorte qu'il reste un collecteur complet a l'heure maximale. Nous dérivons également le non-effondrement pour certaines solutions anciennes à proximité de leur temps maximal. Chacun de ces deux resultats sont liés à la régularité des limites des solutions. Dans le troisième chapître, nous montrons qu'une estimation de type Shi d'ordre un est valable pour tenseur de Ricci sur des variétés qui satisfont l'inégalité Bianchi faibles. Le dernier chapître s'interesse aux gradient solitons de Ricci qui sont en expansion. Nous discutons du problème de classification et montrons que chaque cône tangent à l'infini d'un soliton expansion à "fast-than-quadratic-decay" courbure doit être $mathbb{R}^n$. / This thesis consists of four chapters and an appendix. The first chapter is dedicated to the fundamental ideas of the theory of Ricci flow, which shows how our works are connected to the whole story. In the second chapter, we construct a solution of Ricci flow on a rotationally symmetric manifold such that it remains a complete manifold at the maximal time. We also derive a noncollapsing property for certain ancient solutions near their maximal times. Both of these two results are related to the regularity of limits of solutions. In the third chapter, we show that a first order Shi-type estimate holds for Ricci tensor on manifolds which satisfy the weak Bianchi inequality. The last chapter is concerned with expanding gradient Ricci solitons. There we discuss the classification problem and show that every tangent cone at infinity of an expanding soliton with fast-than-quadratic-decay curvature must be $mathbb{R}^n$.