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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

Every closed 4-manifold with a geometric structure modeled on the Lie group $${\mathrm{Sol}_1}^{4}$$ , or equivalently, infra-solvmanifold of $${\mathrm{Sol}_1}^{4}$$ , is either a mapping torus of an infra-nilmanifold of the 3-dimensional geometry $${\mathrm{Nil}}$$ , or a union of twisted I-bundles over infra-nilmanifolds of $${\mathrm{Nil}}$$ . We determine ab initio presentations for the fundamental groups of the $${\mathrm{Sol}_1}^{4}$$ -geometry manifolds.

C∞-schemes are a generalisation of manifolds that have nice properties such as the existence of fibre products. C∞-schemes have been used as a model for synthetic differential geometry, as in Dubuc, Kock, and Moerdijk and Reyes, and for defining derived differential geometry as in Lurie, and Spivak. Manifolds with corners are a generalisation of manifolds locally modelled on [0,∞)k × Rn-k, and their smooth maps behave well with respect to the corners as in Melrose. In particular, Joyce describes a corner functor from the category of manifolds with corners to the category of 'interior' manifolds with corners with mixed dimension. C∞-algebraic geometry with corners is the study of C∞-rings and C∞-schemes with corners, which we define in this thesis. We define (local/interior/firm) C∞-rings with corners, and study categorical properties such as the existence of limits and colimits using various adjoint functors. We describe a spectrum functor from C∞-rings with corners to local C∞-ringed spaces with corners, and show this a right adjoint to a global sections functor. We define C∞-schemes with corners using this spectrum functor. We show there is a full and faithful embedding of the category of manifolds with corners into the category of firm C∞-schemes with corners, and that fibre products of firm $C^\infty$-schemes with corners exist. We show that manifolds with corners are affine under geometric conditions. We define (b-)cotangent sheaves of C∞-schemes with corners and show they correspond to the (b-)cotangent bundles of manifolds with corners of Joyce. We describe the categories of interior local C∞-ringed spaces with corners and interior firm C∞-schemes with corners. We construct corner functors for both of these categories, which are right adjoint to the inclusion of these interior spaces/schemes into the non-interior ones. We show that these corner functors correspond to the corner functor for manifolds with corners. We expect applications of this work in defining derived spaces with corners in derived differential geometry, and we explore the connections of this work to log geometry and the positive log differentiable spaces of Gillam and Molcho.

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors., Comment: 16 pages, 22 figures

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparating curves to the nonseparating curve graph of the cover span quasiconvex subgraphs which are infinite diameter and not coarsely equal to the nonseparating curve graph of the cover. In the finite type case, we also reprove a theorem of Hamenst\"{a}dt identifying the Gromov boundary with the space of ending laminations on full genus subsurfaces. We introduce several tools based around the analysis of bicorn curves and laminations which may be of independent interest for studying the geometry of nonseparating curve graphs of infinite type surfaces and their boundaries., Comment: Final version to appear in Geometriae Dedicata

In this article, we study the geometry of bisections of certain rational elliptic surfaces. As an application, we give examples of Zariski $$N$$ -plets for conic arrangements, which generalize Namba-Tsuchihashi’s result.

Motivated by the nonexistence of closed spacelike hypersurfaces in the anti-de Sitter space $$\mathbb H_1^{n+1}$$ , in this paper we study the geometry of complete spacelike hypersurfaces immersed in $$\mathbb H_1^{n+1}$$ with either constant mean or scalar curvature. In this setting, under suitable restrictions on the norm of the tangential component of a fixed timelike vector, we show that such hypersurfaces must be isometric to certain hyperbolic spaces. Our approach is based on the use of Bochner’s technique jointly with appropriated generalized maximum principles.

We call a K3 surface \(X\) a 2-elementary K3 surface if the Neron–Severi lattice \(S_X\) satisfies the condition that \(S_X^{*}/S_X\) is a 2-elementary group, where \(S_X^{*}:={{\mathrm{Hom}}}(S_X,\mathbb {Z})\) . In this paper, in the case where \(X\) is a 2-elementary K3 surface given by a double cover of a smooth Del Pezzo surface of degree \(4\le d\le 8\) and \(L\) is a base point free and big line bundle on \(X\), for any smooth curve \(C\) in the linear system \(|L|\), we investigate line bundles on \(C\) which compute the Clifford index of \(C\).

In this text we show that the deformation space of a nodal surface $X$ of degree $d$ is smooth and of the expected dimension if $d\leq 7$ or $d\geq 8$ and $X$ has at most $4d-5$ nodes. (The case $d\leq 7$ was previously covered by Alexandru Dimca by using different techniques.) For $d\geq 8$ we give explicit examples of nodal surfaces with $4d-4$ nodes, for which the tangent space to the deformation space has larger dimension than expected. We give a short discussion on the shape of the deformation space of surfaces of the form $f_1f_2+f_3^2f_4$, where $f_1$ is a linear form., v2: Added a reference to a similar result by Alexandru Dimca and a discussion on the difference between Dimca's result and ours v3: Expanded several arguments

In the paper referred to in the title, we proved that any immersed flat tori in $$S^3$$ whose mean curvature does not change sign has extrinsic diameter $$\pi $$ . Although the main result there is correct, there is a gap of the proof of this fact. The purpose here is to correct the previous paper’s argument and clarify the statement.

In this paper, we consider a kind of area preserving non-local flow for convex curves in the plane. We show that the flow exists globally, the length of evolving curve is non-increasing, and the curve converges to a circle in C^{\infty} sense as time goes into infinity., Comment: 14 pages

The communication processes of knowledge creation represent a particular class of human dynamics where the expertise of individuals plays a substantial role, thus offering a unique possibility to study the structure of knowledge networks from online data. Here, we use the empirical evidence from questions-and-answers in mathematics to analyse the emergence of the network of knowledge contents (or tags) as the individual experts use them in the process. After removing extra edges from the network-associated graph, we apply the methods of algebraic topology of graphs to examine the structure of higher-order combinatorial spaces in networks for four consecutive time intervals. We find that the ranking distributions of the suitably scaled topological dimensions of nodes fall into a unique curve for all time intervals and filtering levels, suggesting a robust architecture of knowledge networks. Moreover, these networks preserve the logical structure of knowledge within emergent communities of nodes, labeled according to a standard mathematical classification scheme. Further, we investigate the appearance of new contents over time and their innovative combinations, which expand the knowledge network. In each network, we identify an innovation channel as a subgraph of triangles and larger simplices to which new tags attach. Our results show that the increasing topological complexity of the innovation channels contributes to networks architecture over different time periods, and is consistent with temporal correlations of the occurrence of new tags. The methodology applies to a wide class of data with the suitable temporal resolution and clearly identified knowledge-content units. Supplementary material: Dataset [https://vinar.vin.bg.ac.rs/handle/123456789/9420] Supplementary material: Dataset [https://vinar.vin.bg.ac.rs/handle/123456789/9421]

Given a convex body \({K\subset\mathbb{R}^n}\) (n ≥ 1) which contains o in its interior and \({{\bf u} \in S^{n-1}}\), we introduce conic volume ratio r(K, u) of K in the direction of u by $$r(K, {\bf u})=\frac{vol(cone(K,{\bf u})\cap B_2^n)}{vol(B_2^n)},$$ where cone(K, u) is the packing cone of K in the direction of u. We prove that if K is an o-symmetric convex body in \({\mathbb{R}^n}\) and r(K, u) is a constant function of u, then K must be a Euclidean ball.

In this paper we prove that there is no nonconstant stable quasi-harmonic sphere from \({({\bf{R}^n}, e^{-|x|^2/2(n-2)}ds_0^2)}\) to N.

We construct geosphere laminations for free groups, which are codimension one analogues of geodesic laminations on surfaces. Other analogues that have been constructed by several authors are dimension-one instead of codimension-one. Our main result is that the space of such laminations is compact. This in turn is based on the result that crossing, in the sense of Scott-Swarup, is an open condition. Our construction is based on Hatcher’s normal form for spheres in the model manifold.

Let M be an orientable irreducible 3-manifold with boundary and let S be a subsurface of ∂M. Suppose every curve in ∂S is disk-busting, i.e. every curve in ∂S intersects every compressing disk in M. We show that either (1) M is an I-bundle of which S is a horizontal boundary component or (2) under a natural projection, the image of the disk complex of M has diameter at most 12 in the curve complex of S.

In this paper, we study surfaces of S3 in the context of Lie sphere geometry. We construct invariants with respect to Lie sphere transformations on the surfaces, which determine the surfaces up to a Lie sphere transformation. Finally we classify completely the homogeneous surfaces in S3 with respect to the Lie sphere transformation group of S3.

Duals of the basic projection and mixed projection inequalities are established for intersection and mixed intersection bodies.

We study 4 × 4-labyrinth fractals, which are self similar dendrites. For all 4 × 4-labyrinth fractals we answer the question, whether there is a curve of finite length in the fractal from one point to another point in the fractal. In the first case, between any two points in the fractal there is a unique arc a, the length of a is infinite, and the set of points, where no tangent exists to a, is dense in a. In the second case, there are also pairs of points between that there is a unique arc of finite length.

We compute Holder Complexes, i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces.

If M is an isoparametric hypersurface in a sphere S n with four distinct principal curvatures, then the principal curvatures κ1, . . . , κ4 can be ordered so that their multiplicities satisfy m 1 = m 2 and m 3 = m 4, and the cross-ratio r of the principal curvatures (the Lie curvature) equals −1. In this paper, we prove that if M is an irreducible connected proper Dupin hypersurface in R n (or S n ) with four distinct principal curvatures with multiplicities m 1 = m 2 ≥ 1 and m 3 = m 4 = 1, and constant Lie curvature r = −1, then M is equivalent by Lie sphere transformation to an isoparametric hypersurface in a sphere. This result remains true if the assumption of irreducibility is replaced by compactness and r is merely assumed to be constant.

The first part of this paper is a survey on Teichmuller curves and Veech groups, with emphasis on the special case of origamis where much stronger tools for the investigation are available than in the general case. In the second part we study a particular configuration of origami curves in genus 3: A “base” curve is intersected by infinitely many “transversal” curves. We determine their Veech groups and the closure of their locus in M 3.

A stable periodic billiard path in a triangle is a billiard path which persists under small perturbations of the triangle. This article gives a geometric proof that no right triangles have stable periodic billiard paths.

We calculate the area of the smallest triangle and the area of the smallest virtual triangle for many known lattice surfaces. We show that our list of the lattice surfaces for which the area of the smallest virtual triangle greater than .05 is complete. In particular, this means that there are no new lattice surfaces for which the area of the smallest virtual triangle is greater than .05. Our method follows an algorithm described by Smillie and Weiss and improves on it in certain respects., Comment: Minor revisions thanks to the suggestions of Anja Randecker and Alex Wright

We present three families of exact, cohomogeneity-one Einstein metrics in $(2n+2)$ dimensions, which are generalizations of the Stenzel construction of Ricci-flat metrics to those with a positive cosmological constant. The first family of solutions are Fubini-Study metrics on the complex projective spaces $CP^{n+1}$, written in a Stenzel form, whose principal orbits are the Stiefel manifolds $V_2(R ^{n+2})=SO(n+2)/SO(n)$ divided by $Z_2$. The second family are also Einstein-K\"ahler metrics, now on the Grassmannian manifolds $G_2(R^{n+3})=SO(n+3)/((SO(n+1)\times SO(2))$, whose principal orbits are the Stiefel manifolds $V_2(R^{n+2})$ (with no $Z_2$ factoring in this case). The third family are Einstein metrics on the product manifolds $S^{n+1}\times S^{n+1}$, and are K\"ahler only for $n=1$. Some of these metrics are believed to play a role in studies of consistent string theory compactifications and in the context of the AdS/CFT correspondence. We also elaborate on the geometric approach to quantum mechanics based on the K\"ahler geometry of Fubini-Study metrics on $CP^{n+1}$, and we apply the formalism to study the quantum entanglement of qubits., Comment: 31 pages

In this work we prove the existence of totally geodesic two-dimensional foliation on the Lorentzian Heisenberg group H3. We determine the Killing vector fields and the Lorentzian geodesics on H3.

A hypersurface f : M → Rn+1 in an affine (n+1)-space is called centroaffine if its position vector is always transversal to f*(TM) in Rn+1. In this paper, we establish a general optimal inequality for definite centroaffine hypersurfaces in Rn+1 involving the Tchebychev vector field. We also completely classify the hypersurfaces which verify the equality case of the inequality.

One of the main tools in the study of real foliations of codimension one is the use of transverse sections. Given a codimension one real foliation on a compact manifold there is always a transverse circle and the topology of its embedding into the ambient manifold often has important consequences on the global behavior of the foliation; as in Haefliger’s Theorem and Novikov’s c7ompact leaf theorem for instance.

Medina and Revoy gave in (Medina and Revoy, Manuscripta Math. 52 (1985) 81–95) a rich class of examples of Lorentzian homogeneous, compact manifolds. These manifolds, the oscillator manifolds, are quotients of the so-called oscillator groups by lattices. Except for the ‘nice’ quotients of SL(2, ℝ), these manifolds are roughly the only Lorentzian, homogeneous manifolds of finite volume, with non-compact group of isometries (Adams and Stuck, Invent. Math. 129 (1997) 239–261; Zhegib, Sur les espaces-temps homoge`nes, Ecole Normale, Lyon, 1995). In this paper we exhibit new locally symmetric structures on the oscillator manifolds. These structures are obtained from explicit pseudo-Riemannian, locally symmetric, left-invariant metrics and from affine structures on the oscillator groups. Other examples of locally symmetric manifolds are provided by the determination of the group of isometries of the oscillator groups.