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In this dissertation, we discuss the behavior of Willmore flow, a fourth-order geometric flow, for complete, properly immersed surfaces in Euclidean space.We develop a-priori estimates for weighted Willmore flows. The estimates are later used to generalize Kuwert and Schäzle’s short-time existence theorem in their 2002 article for complete surfaces with bounded geometry, to find a condition for uniqueness of Willmore flows on complete surfaces, and show gap phenomena of Willmore energy. We also discuss blowups of Willmore flow, as constructed by Kuwert and Schätzle in their 2001 article.We also discuss the Fredholm property of Laplacian operator on the space of normal vector fields, in view of weighted Sobolev spaces as defined by Lockhart in his 1987 article. We give a few conjectures regarding the Fredholm property of linearization of Willmore tensor, Łojasiewicz–Simon inequality, and stability of minimal surfaces with finite energy as Willmore surfaces.

The necessary conditions for a quantization of a module over an algebra on a symplectic manifold to exist are investigated. Considered is a symplectic algebraic variety $M$ with a fixed deformation quantization $\mathcal{O}_{\hbar}$ of its sheaf of regular functions, and a vector bundle $E$ on M with a deformation quantization of order $k$ (as a module over $\mathcal{O}_{\hbar}$). It is found that range of cohomology classes must vanish if this order admits an extension to quantization of order $\ell>k$. For $\ell

We show that if the cochain complex computing Ext groups (in the category of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products., Comment: 31 pages; to appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci

In the first part of this thesis we establish dimension bounds for spaces of polynomial growth solutions to divergence form parabolic partial differential equations with time-dependent coefficients as well as sharp bounds in the case of time-independent coefficients. The second, main part of this thesis is devoted to smooth, harmonic maps of low index, in particular to the classification of such maps into round spheres. Towards this we first study the change of the index of smooth, harmonic submersions into round spheres upon consideration as maps into higher-dimensional round spheres, that is after composition with the totally geodesic inclusion map into a higher-dimensional sphere. This is illustrated by the examples of the Hopf maps $S^3\rightarrow\,S^2$ and $S^1\rightarrow\,S^1$. As an application we can describe for $n\geq3$ a class of harmonic morphisms of low index from the round $n$-sphere into submanifolds of the round $n$-sphere arising from smooth maps of constant rank. Finally we prove an El Soufi-type index bound for smooth, harmonic maps from simply connected Riemannian manifolds satisfying the Killing property into round spheres. This bound depends only on the domain manifold, in particular it does not depend on the dimension of the codomain sphere. Focussing thereafter on the case that the codomain is the round two-sphere, we can show that every smooth, harmonic map of low index with respect to this new index bound must be a harmonic morphism.

This paper provides a well-posedness analysis for a hemivariational inequality of the stationary Navier-Stokes equations by arguments of convex minimization and the Banach fixed point. The hemivariational inequality describes a stationary incompressible fluid flow subject to a nonslip boundary condition and a Clarke subdifferential relation between the total pressure and the normal component of the velocity. Auxiliary Stokes hemivariational inequalities that are useful in proving the solution existence and uniqueness of the Navier–Stokes hemivariational inequality are introduced and analyzed. This treatment naturally leads to a convergent iteration method for solving the Navier–Stokes hemivariational inequality through a sequence of Stokes hemivariational inequalities. Equivalent minimization principles are presented for the auxiliary Stokes hemivariational inequalities which will be useful in developing numerical algorithms.

We initiate the study of the higher-order Escobar constants $I_k(M)$, $k\geq 3$, on bounded planar domains $M$. The Escobar constants $I_k$ of the unit disk and a family of polygons are provided., Comment: 18 pages

We prove that every Jordan curve in $$\mathbb {R}^2$$ inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve.

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a Slice Theorem for Fr\'echet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg-Witten moduli space and the moduli space of pseudoholomorphic curves., Comment: Published version

J-holomorphic curves in nearly Kähler$$\mathbb {CP}^3$$CP3are related to minimal surfaces in$$S^4$$S4as well as associative submanifolds in$$\Lambda ^2_-(S^4)$$Λ-2(S4). We introduce the class of transverseJ-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in$$S^4$$S4and construct moment-type maps from$$\mathbb {CP}^3$$CP3to relate them to the theory of$$\mathrm {U}(1)$$U(1)-invariant minimal surfaces on$$S^4$$S4.

The Riemannian product $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ , where $${\mathbb{M}}_i(c_i)$$ denotes the 2-dimensional space form of constant sectional curvature $$c_i \in {\mathbb{R}}$$ , has two different $${\mathrm{Spin}^{\mathrm{c}}}$$ structures carrying each a parallel spinor. The restriction of these two parallel spinor fields to a 3-dimensional hypersurface M characterizes the isometric immersion of M into $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ . As an application, we prove that totally umbilical hypersurfaces of $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_1(c_1)$$ and totally umbilical hypersurfaces of $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ ( $$c_1 \ne c_2$$ ) having a local structure product are of constant mean curvature.

A complex K3 surface or an algebraic K3 surface in characteristics distinct from $2$ cannot have more than $16$ disjoint nodal curves., A complex K3 surface or an algebraic K3 surface in characteristics distinct from 2 cannot have more than 16 disjoint nodal curves.

Let $\Gamma$ be an $L$-shape arc consisting of 2 line segments that meet at an angle different from $\pi$ in the complex $z$-plane $\CC$. This paper is to investigate the behavior of the polynomial interpolants at the Fej\'er points, defined by $\{z_{n,k} = \psi(e^{i(2k\pi + \theta)/(n+1)})\}$ for any choice of $\theta$. In this regard, we recall that for the interval [-1, 1], the Fej\'er points $\{z_{n,k} = \psi^*(e^{i(2k+1)\pi/(n+1)})\}$ agree with the Chebyshev points and that the Chebyshev points are most commonly used as nodes for Lagrange polynomial interpolation. On the other hand, numerical experimentation demonstrates that for a typical open $L$-shape arc $\Gamma$, the Lebesgue constants tend to $\infty$ at the rate of $O((log(n))^2)$, as the polynomial degree $n$ increases, while the $A_{p}$-weight conditions for the Fej\'er points $\{z_{n,k}\}$ do not carry over from [-1, 1] to a truly $L$-shape arc. Further numerical experiments also demonstrate that the least upper bounds of the Marcinkiewicz-Zygmund inequalities for the canonical Lagrange interpolation polynomials at $\{z_{n,k}\}$ seem to grow at the rate of $n^{\beta}$, for some $\beta >0$ that depends on $p >1$., Comment: 16 pages. 7 figures and 3 tables. Keywords:Marcinkiewicz-Zygmund inequalities; $A_p$ weight; Lebesgue constants. J. Geom. Anal. (2021)

Tallennetaan OA-artikkeli, kun julkaistu / KS Gaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. They furthermore stand as important special cases for frameworks providing geometries for probability measures, as the resulting geometry on Gaussians is often expressible in closed-form under the frameworks. In this work, we study the Gaussian geometry under the entropy-regularized 2-Wasserstein distance, by providing closed-form solutions for the distance and interpolations between elements. Furthermore, we provide a fixed-point characterization of a population barycenter when restricted to the manifold of Gaussians, which allows computations through the fixed-point iteration algorithm. As a consequence, the results yield closed-form expressions for the 2-Sinkhorn divergence. As the geometries change by varying the regularization magnitude, we study the limiting cases of vanishing and infinite magnitudes, reconfirming well-known results on the limits of the Sinkhorn divergence. Finally, we illustrate the resulting geometries with a numerical study.

The object of the paper is to study the generalized Einstein tensor G(X,Y) on almost pseudo-Ricci symmetric manifolds, A(PRS)n. Considering the generalized Einstein tensor G(X,Y) as conservative, cyclic parallel and Codazzi type, it is investigated the properties of such a manifold.

Preface: These are lectures on differential geometry written for undergraduate students at University College Cork in 2017. They assume elementary knowledge of metric spaces and topological spaces, including the concepts of homeomorphism, Hausdorff space, quotient space, and covering space. I recommend the reader on a first reading skip any section that says it can be omitted, and skip all problems marked with a ˚, although it might be worthwhile to read the hints to the ˚ problems.

Preface: To the reader who wants to dip a toe in the water: read chapter 1. The reader who continues on to chapters 2 and 4 will pick up the rest of the tools. Subsequent chapters prove the theorems. We assume that the reader is familiar with elementary differential geometry on manifolds and with differential forms. These lectures explain how to apply the Cartan–Kähler theorem to problems in differential geometry. Given some differential equations, we want to decide if they are locally solvable. The Cartan–Kähler theorem gives a linear algebra test: if the equations pass the test, they are locally solvable. We give the necessary background on partial differential equations in appendices A, B, and the (not so necessary) background on moving frames in appendices D, F, G. The reader should be aware of [4], which we will follow closely, and also the canonical reference work [3] and the canonical textbook [19].

The Following is a Research Thesis by the author of the 8T. The new Infinitytheory aims to present the subject of Infinities as analyzedfrom a sync of set theory and the author forte, Calculus of Variations/Lagrangians. Insert V1.1 - Decays of Infinities - Page 73. The file include three parts, first the Variational approach on Infinities, the second part -classification of Infinitiesand the third andlast part is the Nature of Infinites. Not all theorems are .proven, this will be presented at later versions. The Theory contains four main equations, whichare (1),(1.2) and equation(2) The Thesis is written in a form of self refection, allowing the reader to experience how the ideas varied over time.&nbsp

A feature of biodiversity is the abundance of curves displayed by organs and organisms. Curvature is a widespread, convergent trait that has important ecological and evolutionary implications. In pollination ecology, the curvature of flowers and pollinator mouthparts (e.g. hummingbird bills) along the dorsiventral plane has been associated with specialization, competition, and species co-existence. Six methods have historically been used to measure curvature in pollination systems; we provide a solution to this inconsistency by defining curvature using well-established concepts from differential geometry. Intuitively, curvature is the degree to which a line is not straight, but more formally, it is the rate at which the tangent of a curve changes direction with respect to arc length. Here, we establish a protocol wherein a line is fitted against landmarks placed on an image of a curved organ or organism, then curvature is computed at many points along the fitted line and the sum taken. The protocol is demonstrated by studying the development of nectar spur curvature in the flowering plant genus Epimedium (Berberidaceae). By clarifying the definition of curvature, our aim is to make the language of comparative morphology more precise and broadly applicable to capture other curved structures in nature.

The aim of this work is to derive mathematical descriptions of gravitation. Postulating gravitation as a force field, Newton's law of gravitation is heuristically derived by considering linear differential operators invariant under euclidean isometries and by finding the fundamental solution to Helmholtz equation in three dimensions. Thereafter, the theory of differential geometry is introduced, providing a framework for the subsequent review of gravitation as curvature. Lastly, in the light of Einstein's postulates and equivalence principle, Lovelock's proof of uniqueness of Einstein's field equations is presented.

Kesirli analiz matematiğin teorik kısmında büyük bir öneme sahip olduğu gibi uygulama alanında da büyük bir yer edinmiştir. Bu tez çalışmasında Kesirli Analiz teknikleri kullanılarak eğrilerin geometrisi ele alındı. Bu yapılırken kesirli türev operatörleri içerisinden Caputo kesirli türev operatörü incelendi. Bu çalışmanın amacı bir seviye eğrisinin kesirli mertebeden teğet ve normal vektör alanları sayesinde kesirli türevin geometrik karşılıklarına katkıda bulunmaktadır. Bir seviye eğrisinin kesirli mertebeden teğet ve normal vektör alanı kavramları tanımlanmıştır. Bunların standart kavramlar ile arasındaki ilişkiler incelenmiştir. Fractional analysis has a great importance in the theoretical part of mathematics as well as in the field of application. In this thesis, the geometry of curves will be discussed using Fractional Analysis techniques. While doing this, the Caputo fractional derivative operator will be discussed among the fractional derivative operators. The aim of this study is to contribute to the geometrical equivalents of the fractional derivative thanks to the fractional tangent and normal vector fields of a level curve. The concepts of fractional tangent and normal vector fields of a level curve are defined. The relations between these and standard concepts were examined.

We show that, in dimension at least $4$, the set of locally finite simplicial volumes of oriented connected open manifolds is $[0, \infty]$. Moreover, we consider the case of tame open manifolds and some low-dimensional examples., Comment: 12 pages, 1 figure

There are many excellent books of topology; from the bibliography at the end of this book, the reader might look at [1, 2, 3, 5, 6, 7, 8, 9, 11, 10, 12, 13, 14, 15]. So it might have been wise to avoid writing these notes. They explain what I tried to cover in my 2017 lectures on topology for undergraduate students at University College Cork. I assume the reader is familiar with elementary theory of metric spaces. I will also give a few examples from differential geometry, which the reader can ignore, and which assume familiarity with manifolds and diffeomorphisms.

A Koszul-Vinberg manifold is a manifold $M$ endowed with a pair $(\nabla,h)$ where $\nabla$ is a flat connection and $h$ is a symmetric bivector field satisfying a generalized Codazzi equation. The geometry of such manifolds could be seen as a type of bridge between Poisson geometry and pseudo-Riemannian geometry, as has been highlighted in our previous article [\textit{Contravariant Pseudo-Hessian manifolds and their associated Poisson structures}. \rm{Differential Geometry and its Applications} (2020)]. Our objective here will be to pursue our study by focusing in this setting on submanifolds by taking into account some developments in the theory of Poisson submanifolds.

Funder: Munro-Greaves Bursary for Pure Mathematics, Funder: Engineering and Physical Sciences Research Council; doi: http://dx.doi.org/10.13039/501100000266, Non-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

Different arguments led to supposing that the deep origin of phase transitions has to be identified with suitable topological changes of potential related submanifolds of configuration space of a physical system. An important step forward for this approach was achieved with two theorems stating that, for a wide class of physical systems, phase transitions should necessarily stem from topological changes of energy level submanifolds of the phase space. However, the sufficiency conditions are still a wide open question. In this study, a first important step forward was performed in this direction, in fact, a differential equation was worked out which describes how entropy varies as a function of total energy, and this variation is driven by the total energy dependence of a topology-related quantity of the relevant submanifolds of the phase space. Hence, general conditions can be in principle defined for topology-driven loss of differentiability of the entropy.

Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.