arXiv research

Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.

In this paper, we study the CR submanifolds of maximal CR dimension with flat normal connection of a complex projective space. We first investigate the position of the umbilical normal vector in the normal bundle, especially for the submanifolds of dimension 3. Then as the application, we prove the non-existence of a c…

El Soufi-Ilias' theorem establishes a connection between minimal submanifolds of spheres and extremal metrics for eigenvalues of the Laplace-Beltrami operator. Recently, this connection was used to provide several explicit examples of extremal metrics. We investigate the maximality of these metrics and prove that all o…

New algorithm for Coxeter connections with maximally ramified singularities.

This report aims at giving a general overview on the classification of the maximal subgroups of compact Lie groups (not necessarily connected). In the first part, it is shown that these fall naturally into three types: (1) those of trivial type, which are simply defined as inverse images of maximal subgroups of the cor…

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…

We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.

The paper describes and analyzes maximal Lorentzian surfaces with a Killing field, proving completeness criteria.

The paper parametrizes spaces of maximal framed representations for a specific type of surface group.

For a fixed compact Riemann surface X, of genus at least 2, we count the number of connected components of the moduli space of maximal Higgs bundles over X for the hermitian groups $PSp(2n,R)$, $PSO^*(2n)$, $PSO_0(2,n)$ and $E_6^{-14}$. Hence the same result follows for the number of connected components of the moduli …

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ with an expli…

The paper studies connections in superintegrable systems, revealing geometric insights.

Given a smooth simply connected planar domain, the area is bounded away from zero in terms of the maximal curvature alone. We show that in higher dimensions this is not true, and for a given maximal mean curvature we provide smooth embeddings of the ball with arbitrary small volume.

Upper bounds on map degrees for various manifold types.

Study maximal antipodal sets in exceptional symmetric spaces.

A geometric framework for metrics of maximal acceleration which is applicable to large proper accelerations is discussed, including a theory of connections associated with the geometry of maximal acceleration. In such a framework it is shown that the uniform bound on the proper maximal acceleration implies an uniform b…

Solves Deligne-Simpson problem for special connections on Gm.

In this work, it is shown that a simply-connected, rationally-elliptic torus orbifold is equivariantly rationally homotopy equivalent to the quotient of a product of spheres by an almost-free, linear torus action, where this torus has rank equal to the number of odd-dimensional spherical factors in the product. As an a…

Study shows almost complex structures with certain tensor properties are prevalent.

We study in this paper the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for …

We show that any effective isometric torus action of maximal rank on a compact Riemannian manifold with positive (sectional) curvature and maximal symmetry rank, that is, on a positively curved sphere, lens space, complex or real projective space, is equivariantaly diffeomorphic to a linear action. We show that a compa…

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli …

We extend the equivariant classification results of Escher and Searle for closed, simply connected, non-negatively curved Riemannian $n$-manifolds admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove…

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

The paper explores reflection principles for lightlike line segments on maximal surfaces.

Study feature representations induced by dependence between variables.

Kernel methods linked to feature subspaces and maximal correlation kernels.

Maximizing margins leads to lossless compression of training data.

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples $(Δ, \mathfrak{h}, G)$ of n…

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine the group structures of the fixed points subgroup. They correspond to the classif…

The paper examines properties of self-affine Sierpiński sponges using metric invariants.

Let $G$ be a compact connected Lie group and $H$ a closed subgroup of $G$. Suppose the homogeneous space $G/H$ is effective and has dimension 3 or higher. Consider a $G$-invariant, symmetric, positive-semidefinite, nonzero (0,2)-tensor field $T$ on $G/H$. Assume that $H$ is a maximal connected Lie subgroup of $G$. We p…

In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group $G= F_4, E_6, E_7$. We shall find involutive automorphisms of $G$ such that the connected component of the intersection of those fixed point subgroups coincides with the maximal torus of $G$.

We present a family of expectation-maximization (EM) algorithms for binary and negative-binomial logistic regression, drawing a sharp connection with the variational-Bayes algorithm of Jaakkola and Jordan (2000). Indeed, our results allow a version of this variational-Bayes approach to be re-interpreted as a true EM al…

We classify closed, simply connected $n$-manifolds of non-negative sectional curvature admitting an isometric torus action of maximal symmetry rank in dimensions $2\leq n\leq 6$. In dimensions $3k$, $k=1,2$ there is only one such manifold and it is diffeomorphic to the product of $k$ copies of the 3-sphere.

The curvature of web curves is studied in 3D manifolds.

A spacetime can be embedded in an enveloping space with all its extensions.

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M…

Any 7-dimensional cocalibrated G_2-manifold admits a unique connection $\nabla$ with skew symmetric torsion. We study these manifolds under the additional condition that the $\nabla$-Ricci tensor vanishes. In particular, we describe their geometry in case of a maximal number of $\nabla$-parallel vector fields.

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m>4) of positive section…

Optimizes eigenvalues on surfaces with symmetries.

We develop the theory of maximal representations of the fundamental group of a compact connected oriented surface with boundary, into a group of Hermitian type. For any such representation we define the Toledo invariant, for which we establish properties such as uniform boundedness on the representation variety, additi…

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the…

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by…

The paper studies the holonomy of spherically symmetric Finsler metrics.

We prove that for a compact subgroup $H$ of an almost connected locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $H$ is a maximal compact subgroup of $G$, (2) $G/H$ is contractible, (3) $G/H$ is homeomorphic to a Euclidean space, (4) $G/H$ is an AE for paracompact spaces, (5) $…

We classify closed, simply-connected non-negatively curved 5-manifolds admitting an (almost) effective, isometric $T^3$ or $T^2$ action. As a direct consequence, we show that for any manifold, of dimensions up to and including 9 under the same hypotheses, the maximal symmetry rank is equal to $[2n/3]$ and the free rank…