Search results

Title: Long-time convergence of harmonic map heat flows from surfaces into Riemannian manifolds
Creator: Choi, Kwangho
Date: 2011
Collection: Electronic Theses & Dissertations
Description: We study the long-time convergence of harmonic map heat flows from a closed Riemann surface into a compact Riemannian manifold. P. Topping constructed an example of a flow that does not converge in the infinite-time limit. Motivated by the observation that Topping's flow has accumulation points at which the Hessian of the energy function is degenerate, we prove convergence under the assumptions that (a) the Hessian of the energy at an accumulation point is positive definite, and (b) no...
Show moreWe study the long-time convergence of harmonic map heat flows from a closed Riemann surface into a compact Riemannian manifold. P. Topping constructed an example of a flow that does not converge in the infinite-time limit. Motivated by the observation that Topping's flow has accumulation points at which the Hessian of the energy function is degenerate, we prove convergence under the assumptions that (a) the Hessian of the energy at an accumulation point is positive definite, and (b) no bubbling occurs at infinite time. In addition, we present examples of heat flows for geodesics which show that the convexity of the energy function and convergence at infinite time may not hold even for 1-dimensional harmonic map heat flows.
Show less

Title: Jones-type link invariants and applications to 3-manifold topology
Creator: Lee, Christine Ruey Shan
Date: 2015
Collection: Electronic Theses & Dissertations
Description: "It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semi-adequate link has a well-defined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semi-adequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new...
Show more"It is known that the Slope Conjecture is true for an adequate link, and that the colored Jones polynomial of a semi-adequate link has a well-defined tail (head) consisting of stable coefficients, which carry geometric and topological information of the link complement. We study the colored Jones polynomial of a link that is not semi-adequate and show that a tail (head) consisting of stable coefficients of the polynomial can also be defined. Then, we prove the Slope Conjecture for a new family of pretzel knots which are not adequate. We also study the relationship between the Jones polynomial and the topology of the knot complement by relating its coefficients to the non-orientable genus of an alternating knot. The two-sided bound we obtain can often determine the non-orientable genus. Lastly, we develop the connection between the Jones polynomial and a knot invariant coming from Heegaard Floer homology by using it to classify 3-braids which are L-space knots." -- Abstract.
Show less

Title: Statistical properties of some almost Anosov systems
Creator: Zhang, Xu
Date: 2016
Collection: Electronic Theses & Dissertations
Description: We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai-Ruelle-Bowen measures (SRB measures), where the almost Anosov diffeomorphism is a system which is hyperbolic everywhere except for one point. At the indifferent fixed point, the Jacobian matrix is an identity matrix. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the...
Show moreWe investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai-Ruelle-Bowen measures (SRB measures), where the almost Anosov diffeomorphism is a system which is hyperbolic everywhere except for one point. At the indifferent fixed point, the Jacobian matrix is an identity matrix. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and can be expressed by using coefficients of the third order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed points.We discuss the relationship between the existence of SRB measures and the differentia- bility of some almost Anosov diffeomorphisms near the indifferent fixed points in dimensions bigger than one. The eigenvalue of Jacobian matrix at the indifferent fixed point along the one-dimensional contraction subspace is less than one, while the other eigenvalues along the expansion subspaces are equal to one. As a consequence, there are twice-differentiable al- most Anosov diffeomorphisms that admit infinite SRB measures in two or three-dimensional spaces; there exist twice-differentiable almost Anosov diffeomorphisms with SRB measures in dimensions bigger than three. Further, we obtain the polynomial lower and upper bounds for the correlation functions of these almost Anosov maps that admit SRB measures.
Show less

Title: A combinatorial approach to knot theory : volume bounds for hyperbolic semi-adequate link complements
Creator: Giambrone, Adam Joseph
Date: 2014
Collection: Electronic Theses & Dissertations
Description: An interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semi-adequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semi-adequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed...
Show moreAn interesting goal in knot theory is to discover how much geometric information about a link can be carried by a representative projection diagram of that link. To this end, we show that the volumes of certain hyperbolic semi-adequate links can be bounded above and below in terms of two diagrammatic quantities: the twist number and the number of special tangles in a semi-adequate diagram of the link. Given this result, we then narrow our focus to families of plat closures, families of closed braids, and families of links that have both plat and closed braid aspects. By more closely studying each of these families, we can often improve the lower bounds on volume provided by the main result. Furthermore, we show that the bounds on volume can be expressed in terms of a single stable coefficient of the colored Jones polynomial. By doing this, we provide new collections of links that satisfy a Coarse Volume Conjecture. The main approach of this entire work is to use a combinatorial perspective to study the connections among knot theory, hyperbolic geometry, and graph theory.
Show less

Title: Topics in knot theory : on generalized crossing changes and the additivity of the Turaev genus
Creator: Balm, Cheryl Lyn Jaeger
Date: 2013
Collection: Electronic Theses & Dissertations
Description: We first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev...
Show moreWe first study cosmetic crossing changes and cosmetic generalized crossing changes in knots of genus one, satellite knots, and knots obtained via twisting operations on standardly embedded tori in the knot complement. As a result, we find obstructions to the existence of cosmetic generalized crossing changes in several large families of knots. We then study Turaev surfaces and use decomposing spheres to analyze the additivity of the Turaev genus for the summands of composite knots with Turaev genus one.
Show less

Title: The Homology Polynomial and the Burau Representation for Pseudo-Anosov Braids
Creator: Shultz, Warren Michael
Date: 2021
Collection: Electronic Theses & Dissertations
Description: The homology polynomial is an invariant for pseudo-Anosov mapping classes \cite{Birman2010}. We study the homology polynomial as an invariant for pseudo-Anosov braids and its connection to the Burau representation. Given a pseudo-Anosov braid $\beta\in B_n$, we determine necessary and sufficient conditions under which the homology polynomial of $\beta$ is equal to the the characteristic polynomial of the image of $\beta$ under the Burau representation. In particular, we build upon \cite...
Show moreThe homology polynomial is an invariant for pseudo-Anosov mapping classes \cite{Birman2010}. We study the homology polynomial as an invariant for pseudo-Anosov braids and its connection to the Burau representation. Given a pseudo-Anosov braid $\beta\in B_n$, we determine necessary and sufficient conditions under which the homology polynomial of $\beta$ is equal to the the characteristic polynomial of the image of $\beta$ under the Burau representation. In particular, we build upon \cite{Band2007} and show that the orientation cover associated to a pseudo-Anosov braid is equivalent to a quotient to the Burau cover when the measured foliations associated to $\beta$ have odd-ordered singularities at each puncture and any singularity that occurs in the interior of $D_n$ is even-ordered. We next construct an algorithm which allows us to determine the homology polynomial from the Burau representation for an arbitrary pseudo-Anosov braid. As an application, we show how to easily determine the homology polynomial for large family of pseudo-Anosov braids.
Show less

Title: Three Variations on Johnson-Lindenstrauss Maps for Submanifolds of Euclidean Space via Reach
Creator: Tavakoli, Arman
Date: 2021
Collection: Electronic Theses & Dissertations
Description: In this thesis we investigate 3 variations of the classical Johnson-Lindenstrauss (JL) maps. In one direction we build on the earlier work of Wakin and Eftekhari (2015), by considering generalizations to manifolds with boundary. In a second direction we extend the work of Noga Alon (2003) for lower bounds for the final embedding dimension in JL maps. In the third direction, we consider matrices with fast matrix-vector multiply and improve the run-time in the earlier work of Oymak, Recht and...
Show moreIn this thesis we investigate 3 variations of the classical Johnson-Lindenstrauss (JL) maps. In one direction we build on the earlier work of Wakin and Eftekhari (2015), by considering generalizations to manifolds with boundary. In a second direction we extend the work of Noga Alon (2003) for lower bounds for the final embedding dimension in JL maps. In the third direction, we consider matrices with fast matrix-vector multiply and improve the run-time in the earlier work of Oymak, Recht and Soltanolkotabi (2018), and Ailon and Liberty (2009).This thesis is organized into 6 chapters. The three variations are discussed in chapters 4, 5 and 6. The variation for manifolds with boundary is presented in chapter 4. The lower bound problem is discussed in chapter 5, and chapter 6 is regarding the run-time improvements. The first chapter is an introduction to Johnson-Lindenstrauss maps. The second chapter is about a regularity parameter called reach and geometrical estimates for manifolds. The third chapter is regarding two geometry questions about reach that arise from the discussions in chapter 2.
Show less

Title: Some computations and applications of Heegaard Floer correction terms
Creator: Park, Kyungbae
Date: 2014
Collection: Electronic Theses & Dissertations
Description: In this dissertation we study some computations and applications of Heegaard Floer correction terms. In particular we explore the correction terms for the double covers of the three-sphere branched along the Whitehead doubles of knots. As a consequence we show that Whitehead double and iterated double of some classes of knots are independent in the smooth knot concordance group. We also compute the correction terms of non-trivial circle bundles over oriented surfaces and discuss how they can...
Show moreIn this dissertation we study some computations and applications of Heegaard Floer correction terms. In particular we explore the correction terms for the double covers of the three-sphere branched along the Whitehead doubles of knots. As a consequence we show that Whitehead double and iterated double of some classes of knots are independent in the smooth knot concordance group. We also compute the correction terms of non-trivial circle bundles over oriented surfaces and discuss how they can be applied to four-dimensional topology.
Show less

Title: Discrete de Rham-Hodge Theory
Creator: Zhao, Rundong
Date: 2020
Collection: Electronic Theses & Dissertations
Description: We present a systematic treatment to 3D shape analysis based on the well-established de Rham-Hodge theory in differential geometry and topology. The computational tools we developed are widely applicable to research areas such as computer graphics, computer vision, and computational biology. We extensively tested it in the context of 3D structure analysis of biological macromolecules to demonstrate the efficacy and efficiency of our method in potential applications. Our contributions are...
Show moreWe present a systematic treatment to 3D shape analysis based on the well-established de Rham-Hodge theory in differential geometry and topology. The computational tools we developed are widely applicable to research areas such as computer graphics, computer vision, and computational biology. We extensively tested it in the context of 3D structure analysis of biological macromolecules to demonstrate the efficacy and efficiency of our method in potential applications. Our contributions are summarized in the following aspects. First, we present a compendium of discrete Hodge decompositions of vector fields, which provides the primary building block of the de Rham-Hodge theory for computations performed on the commonly used tetrahedral meshes embedded in the 3D Euclidean space. Second, we present a real-world application of the above computational tool to 3D shape analysis on biological macromolecules. Finally, we extend the above method to an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. Our work on the decomposition of vector fields, spectral shape analysis on static shapes, and evolving shapes has already shown its effectiveness in biomolecular applications and will lead to a rich set of features for machine learning-based shape analysis currently under development.
Show less

Title: Bubble tree construction for Harmonic maps using Deligne-Mumford moduli space
Creator: Park, Woongbae
Date: 2021
Collection: Electronic Theses & Dissertations
Description: We formulate and prove a general compactness theorem for harmonic maps.Convergence is defined using Deligne-Mumford space and families of curves. Given a sequence of harmonic maps from a sequence of closed Riemann surfaces to a compact Riemannian manifold with uniformly bounded energy, the main theorem shows that there is a family of curves and a subsequence such that both the domains and the maps converge off the set of ``non-regular'' nodes. This convergence result extends existing bubble...
Show moreWe formulate and prove a general compactness theorem for harmonic maps.Convergence is defined using Deligne-Mumford space and families of curves. Given a sequence of harmonic maps from a sequence of closed Riemann surfaces to a compact Riemannian manifold with uniformly bounded energy, the main theorem shows that there is a family of curves and a subsequence such that both the domains and the maps converge off the set of ``non-regular'' nodes. This convergence result extends existing bubble-tree construction to the case of varying domains.
Show less

Title: Volumes, determinants, and meridian lengths of hyperbolic links
Creator: Burton, Stephan D., 1987-
Date: 2017
Collection: Electronic Theses & Dissertations
Description: We study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of non-alternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi...
Show moreWe study relationships between link diagrams and link invariants arising from hyperbolic geometry. The volume density of a hyperbolic link K is defined to be the ratio of the hyperbolic volume of K to the crossing number of K. We show that there are sequences of non-alternating links with volume density approaching v_8, where v_8 is the volume of the regular ideal hyperbolic octahedron. We show that the set of volume densities is dense in [0,v_8]. The determinant density of a link K is 2 pi log det(K)/c(K). We prove that the closure of the set of determinant densities contains the set [0, v_8]. We examine the conjecture, due to Champanerkar, Kofman, and Purcell that vol(K) < 2 pi log det (K) for alternating hyperbolic links, where vol(K) = vol(S^3\ K) is the hyperbolic volume and det(K) is the determinant of K. We prove that the conjecture holds for 2-bridge links, alternating 3-braids, and various other infinite families. We show the conjecture holds for highly twisted links and quantify this by showing the conjecture holds when the crossing number of K exceeds some function of the twist number of K.We derive bounds on the length of the meridian and the cusp volumeof hyperbolic knots in terms of the topology of essential surfaces spanned by the knot.We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound.As applications we find knot diagrammatic upper bounds on the meridian length and the cusp volume of hyperbolic adequate knots and we obtain new large families of knots withmeridian lengths bounded above by four. We also discuss applications of our results to Dehn surgery.
Show less

Title: Topics in link homology
Creator: Jaeger, Thomas Constantin
Date: 2011
Collection: Electronic Theses & Dissertations
Description: We prove two results about mutation invariance of link homology theories: Weshow that Khovanov's universal sl(2) homology is invariant under mutationand that the reduced sl(n) homology defined by Khovanov and Rozansky isinvariant under component-preserving positive mutation when n is odd. Wealso give a relationship between the Khovanov homology of a closed positive3-braid and the Khovanov homology of the braid after adding a number of fulltwists.

Title: Asymptotics of the Turaev-Viro invariants and their connections in low-dimensional topology
Creator: Kumar, Sanjay Lakshman
Date: 2021
Collection: Electronic Theses & Dissertations
Description: We study the Turaev-Viro invariants of 3-manifolds as well as their relationship to invariants arising from hyperbolic geometry. We first construct a closed formula for the Turaev-Viro invariants for hyperbolic once-punctured torus bundles. We then examine a conjecture by Chen and Yang which states that the asymptotics of the Turaev-Viro invariants recover the hyperbolic volume of the manifold. Using topological tools, we are able to construct infinite families of new examples of hyperbolic...
Show moreWe study the Turaev-Viro invariants of 3-manifolds as well as their relationship to invariants arising from hyperbolic geometry. We first construct a closed formula for the Turaev-Viro invariants for hyperbolic once-punctured torus bundles. We then examine a conjecture by Chen and Yang which states that the asymptotics of the Turaev-Viro invariants recover the hyperbolic volume of the manifold. Using topological tools, we are able to construct infinite families of new examples of hyperbolic links in the 3-sphere which satisfy the conjecture. Additionally, we show a general method for augmenting a link such that the resulting link has hyperbolic complement which satisfies the conjecture. As an application of the constructed links in the 3-sphere, we extend the class of known examples which satisfy a conjecture by Andersen, Masbaum, and Ueno relating the quantum representations of surface mapping class groups to its Nielsen-Thurston classification. Most notably, we provide explicit elements in the mapping class group for a genus zero surface with n boundary components, for n > 3, as well as elements in the mapping class group for any genus g surface with four boundary components such that the obtained elements satisfy the conjecture. This is accomplished by utilizing the intrinsic relationship between the Turaev-Viro invariants and the quantum representations shown by Detcherry and Kalfagianni.
Show less

Title: ON A FAMILY OF INTEGRAL OPERATORS ON THE BALL
Creator: Tian, Wenchuan
Date: 2021
Collection: Electronic Theses & Dissertations
Description: In this dissertation, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and identify the extremal functions using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally...
Show moreIn this dissertation, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and identify the extremal functions using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.
Show less

Title: Diagrammatic and Geometric Invariants of Hyperbolic Weakly Generalized Alternating Knots
Creator: Bavier, Brandon
Date: 2021
Collection: Electronic Theses & Dissertations
Description: We study the relationship between knot and link diagrams on surfaces and their invariants coming from hyperbolic geometry. A \indef{link diagram on a surface}, denoted $\pi(L)\subset F$, is a way of projecting a link $L$ in some manifold $M$ onto some surface $F\subset M$. This generalizes the notion of a link diagram, and has been studied with a variety of conditions. We will work with the weakly generalized alternating knots and links of Howie and Purcell, which gives certain criteria to...
Show moreWe study the relationship between knot and link diagrams on surfaces and their invariants coming from hyperbolic geometry. A \indef{link diagram on a surface}, denoted $\pi(L)\subset F$, is a way of projecting a link $L$ in some manifold $M$ onto some surface $F\subset M$. This generalizes the notion of a link diagram, and has been studied with a variety of conditions. We will work with the weakly generalized alternating knots and links of Howie and Purcell, which gives certain criteria to ensure the diagram is interesting. A cusp of a hyperbolic knot $K$ is a neighborhood of $K$ in $M\setminus K$, and the cusp volume is the Euclidean volume of a maximal cusp. We show that the cusp volume of a weakly generalized alternating knot, with some additional conditions, is bounded both above and below based on the twist number of $\pi(K)$ and $\chi(F)$. This is done by constructing a new essential surface for $K$ that has nice properties, including having Euler characteristic based on the twist number as opposed to the crossing number. Our bound on cusp volume leads to interesting bounds on other geometric properties of $K$, including slope length and volumes of Dehn surgery. The volume}of a hyperbolic link $L$ in a manifold $M$ is the hyperbolic volume of the complement $M\setminus L$. We can show that volume is also bounded below by the twist number of $\pi(L)$ and $\chi(F)$. We do this by generalizing the Jones polynomial to weakly generalized alternating links, and showing that there is a relation between this polynomial and the twist number, and this polynomial and the volume. Through the course of proving this bound, we also get relations between the guts of the checkerboard surfaces of $L$ and this generalized Jones polynomial. In addition, if we are working inside a thickened surface, the twist number becomes an isotopy invariant of $L$.
Show less

Title: Heegaard Floer homology and L-space knots
Creator: Vafaee, Faramarz
Date: 2014
Collection: Electronic Theses & Dissertations
Description: Heegaard Floer theory consists of a set of invariants of three- and four-dimensional manifolds. Three-manifolds with the simplest Heegaard Floer invariants are called L-spaces, and the name stems from the fact that lens spaces are L-spaces. The overarching goal of the dissertation is to understand L-spaces better. More specifically, this dissertation could be considered as a step towards finding topological characterizations of L-spaces and L-space knots without referencing Heegaard Floer...
Show moreHeegaard Floer theory consists of a set of invariants of three- and four-dimensional manifolds. Three-manifolds with the simplest Heegaard Floer invariants are called L-spaces, and the name stems from the fact that lens spaces are L-spaces. The overarching goal of the dissertation is to understand L-spaces better. More specifically, this dissertation could be considered as a step towards finding topological characterizations of L-spaces and L-space knots without referencing Heegaard Floer homology. We study knots in $S^3$ that admit positive L-space Dehn surgeries. In particular, we give new examples of knots in $S^3$ within both the families of hyperbolic and satellite knots admitting L-space surgeries. It should be pointed out that for satellite knot examples, we use Berge-Gabai knots (i.e. knots in $S^1 \times D^2$ with non-trivial solid torus Dehn surgeries) as the pattern. Moreover, we study the relationship between satellite knots and L-space surgeries in the general setting, i.e. when the pattern is an arbitrary knot in $S^1 \times D^2$.
Show less

Title: The reduced knot Floer complex
Creator: Krcatovich, David Thaddeus
Date: 2014
Collection: Electronic Theses & Dissertations
Description: We define a “reduced” version of the knot Floer complex CFK-(K), and show that itbehaves well under connected sums and retains enough information to compute HeegaardFloer d-invariants of manifolds arising as surgeries on the knot K. As an application toconnected sums, we prove that if a knot in the three-sphere admits an L-space surgery, itmust be a prime knot. As an application to the computation of d-invariants, we show thatthe Alexander polynomial is a concordance...
Show moreWe define a “reduced” version of the knot Floer complex CFK-(K), and show that itbehaves well under connected sums and retains enough information to compute HeegaardFloer d-invariants of manifolds arising as surgeries on the knot K. As an application toconnected sums, we prove that if a knot in the three-sphere admits an L-space surgery, itmust be a prime knot. As an application to the computation of d-invariants, we show thatthe Alexander polynomial is a concordance invariant within the class of L-space knots, andshow the four-genus bound given by the d-invariant of +1-surgery is independent of the genusbounds given by the Ozsvath-Szabo τ invariant, the knot signature and the Rasmussen sinvariant.
Show less

Title: Knot theory of Morse-Bott critical loci
Creator: Ozsarfati, Metin
Date: 2018
Collection: Electronic Theses & Dissertations
Description: We give an alternative proof of that a critical knot of a Morse-Bott function f : S3 → R is a graph knot where the critical set of f is a link in S3. Our proof inducts on the number of index-1 critical knots of f. -- Abstract.

Title: Obstruction and existence for twisted Kähler-Einstein metrics and convexity
Creator: Rao, Ambar
Date: 2013
Collection: Electronic Theses & Dissertations
Description: Let $L \to X$ be an ample holomorphic line bundle over a compact K"{a}hler manifold $(X,omega_{0})$ so that $c_1(L)$ is represented by the K"{a}hler form $\omega_0$. Given a semi-positive real $(1,1)$ form $\eta$ representing $-c_{1}(K_{X}\otimes L)$, one can ask whether there exists a K"ahler metric $\omega\in c_{1}(L)$ that solves the equation $Ric(\omega) -\omega=\eta$. We study this problem by twisting the K"ahler-Ricci flow by $eta$ , that is evolve along the flow $\dot{\omega_t}=\omega_...
Show moreLet $L \to X$ be an ample holomorphic line bundle over a compact K"{a}hler manifold $(X,omega_{0})$ so that $c_1(L)$ is represented by the K"{a}hler form $\omega_0$. Given a semi-positive real $(1,1)$ form $\eta$ representing $-c_{1}(K_{X}\otimes L)$, one can ask whether there exists a K"ahler metric $\omega\in c_{1}(L)$ that solves the equation $Ric(\omega) -\omega=\eta$. We study this problem by twisting the K"ahler-Ricci flow by $eta$ , that is evolve along the flow $\dot{\omega_t}=\omega_{t}+\eta -Ric(\omega_{t})$ starting at $\omega_{0}$. We prove that such a metric exists provided $\omega_{t}^{n}\geq K \omega^{n}_{0}$ for some $K>0$ and all $t \geq 0$. We also study a twisted version of Futaki's invariant, which we show is well-defined if $\eta$ is annihilated under the infinitesimal action of $\eta(X)$. Finally, using Chens $\epsilon$-geodesics instead, we give another proof of the convexity of $\mathcal{L}_{\omega}$ along geodesics, which plays a central to Berman's proof of the uniqueness of critical points of $\mathcal{F}_{\omega}$.
Show less

Title: Stability of the Almost Hermitian Curvature flow
Creator: Smith, D. J. (Daniel J.)
Date: 2013
Collection: Electronic Theses & Dissertations
Description: The Almost Hermitian Curvature flow was introduced by Streets and Tian in order to study almost hermitian structures, with a particular interest in symplectic structures. This flow is given by a diffusion-reaction equation. Hence it is natural to ask the following: which almost hermitian structures are dynamically stable? We show that Calabi-Yau structures as well as Kahler-Einstein structures with first Chern class negative are dynamically stable.

MSU Libraries

Digital Repository

Enabled Filters

Pages

Pages

MSU Libraries

Digital Repository

You are here

Search results

Pages

Pages