arXiv research

We discuss two kinds of almost contact metric structures on a one-parameter family of totally umbilical hyperspheres in the nearly Kaehler unit 6-sphere.

New families of non-singular geodesic orbit nilmanifolds discovered.

We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be…

Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length …

In this paper, we discuss a one parameter family of complex Born-Infeld solitons arising from a one parameter family of minimal surfaces. The process enables us to generate a new solution of the B-I equation from a given complex solution of a special type (which are abundant). We illustrate this with many examples. We …

In this paper, we deduce some rigidity results in warped product spaces under normal variations of CMC hypersurfaces. In particular, we prove the existence of one-parameter families locally rigid on the spatial fiber of Anti-de Sitter Schwarzschild spacetime and one-parameter families with bifurcation points on the spa…

We study geodesics of the form $γ(t)=π(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $π:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of $G$ (i.e. $γ(t)=π(\exp (tX))$, $X\in …

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions fo…

In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same s…

We construct two one-parameter families of minimal properly embedded surfaces in the Lie group Sol3 using a Weierstrass-type representation. These surfaces are not invariant by a one-parameter group of ambient isometries. The first one can be viewed as a family of helicoids, and the second one is a family of minimal an…

Diffeomorphism freedom induces a gauge dependence in the theory of spacetime perturbations. We derive a compact formula for gauge transformations of perturbations of arbitrary order. To this end, we develop the theory of Taylor expansions for one-parameter families (not necessarily groups) of diffeomorphisms. First, we…

The aim of this short note is to announce the existence of a one-parameter family of left-invariant metrics on $S^3$ admitting WK-spinors. This family contains the two non-Einstein Sasakian metrics with WK-spinors on $S^3$, but does not contain the standard sphere $S^3$ with Killing spinors. Moreover, any simply-connec…

The study examines discrete states in hyperbolic spaces using specific transformations.

Constructs non-abelian G2-instantons on ALC members of B7 family.

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric.…

For a smooth family of exact forms on a smooth manifold, an algorithm for computing a primitive family smoothly dependent on parameters is given. The algorithm is presented in the context of a diagram chasing argument in the Čech-de Rham complex. In addition, explicit formulas for such primitive family are presented.

New families of weakly symmetric nilmanifolds discovered.

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further…

New Einstein solvmanifolds created from non-flat Ricci solitons.

New magnetic flow rigidity theorem for negative curvatures.

Develops a new family of signature-changing models on metric manifolds.

Overview of integrable systems with symmetries, focusing on toric and semitoric systems.

In this note, we extend the theory of Chern-Cheeger-Simons to construct canonical invariants for a one-parameter family of flat connections on a smooth manifold. These invariants lie in degrees $(2p-2)$-cohomology with $\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in a variation of paths (p…

Given a closed real analytic Riemannian manifold, we construct and study a one parameter family of adapted complex structures on the manifold of its geodesics.

One-parameter smooth families of circles in the complex plane with the following property are described: a function is polyanalytic if and only if it has meromorphic extension inside any circle from the family, with the only singularity-a pole at the center.

We compute the Chekanov-Eliashberg contact homology of what we call the Legendrian closure of a positive braid. We also construct an augmentation for each such link diagram. Then we apply the monodromy techniques established in an earlier paper to a certain natural loop in the space L' of positive Legendrian (p,q) toru…

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our theorem with a classical result of Salamon and Zehnder. Finally, we use the index theo…

The exterior algebra of a vector space admits a family of braided Hopf structures.

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of…

Constructs a family of genus three minimal surfaces with parallel ends.

This article considers a one-parameter family of circles F_C, which has the interesting property that the null isocline of the family is the largest member of the family. This family of circles is bounded and we consider the problem of deriving an equation for the envelope of F_C. We provide one standard solution, and …

We analyse the fine convergence properties of one parameter families of hyperbolic metrics, on a fixed underlying surface, that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms.

We consider families of Dirac operators on the unit interval which depend on parameters via boundary conditions. We study the associated eta forms and Maslov cocyles. With this simple example we show how previous results of Lesch/Woiciechowski and the first author on the eta invariant of cylinders generalize to the fam…

In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang-Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) w…

New one-parameter families of $SU(2)^2$-invariant instantons found on Calabi-Yau 3-folds.

On a Riemannian manifold we define a one-parameter family of Laplacians acting on sections of any bundle associated to the principal frame bundle via a representation, and show how various examples fit into this framework.

The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method's construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn…

We formulate a correspondence between affine and projective special Kähler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special Kähler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special Kähler manifolds in t…

We show that Scherk's first surface, a one-parameter family of solutions to the minimal surface equation, may be written as a linear superposition of other solutions with specific parametric values.

Develops a new non-abelian framework for Riemann surfaces and differential equations.

Sloth predicts LLM performance using latent skills across families.

We construct one-parameter families of solutions to the Einstein--Klein--Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-p…

A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive secti…

New strategy identifies best Markovian arm with fixed confidence.

An efficient algorithm for Riemannian logarithm on Stiefel manifold family.

The paper discovers the family of identically-derived Euclidean one-parameter even-dimensional differential linear operators with unique eigenproperties, which prove to be inherently related to the emergent characterizations of fundamental building blocks of embedded minimal surfaces and the Nitsche conjecture proof.

We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk second surface and Hoffman-Wohlgemuth example as limit-members.

The paper solves CR curvature prescription on pseudo-Einstein 3-manifolds.