arXiv research

The paper studies geometric loci and their invariants in complex dynamics.

We calculate the Masur-Veech volume of the gothic locus $\mathcal{G}$ in the stratum $\mathcal{H}(2^{3})$ of genus four. Our method is based on the use of the formulae for the Euler characteristics of gothic Teichmüller curves to determine the number of lattice points of given area. We also use this method to recalcula…

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

We compute the Euler characteristics of the recently discovered series of Gothic Teichmüller curves. The main tool is the construction of 'Gothic' Hilbert modular forms vanishing at the images of these Teichmüller curves. Contrary to all previously known examples, the Euler characteristic is not proportional to the Eul…

Study the geometry of lightlike loci on mixed type surfaces in Lorentz-Minkowski 3-space.

The aim of this article is to generalize the notion of the cut locus and to get the structure theorem for it. For this purpose, we first introduce a class of 1-Lipschitz functions, each member of which is called an {\it almost distance function}. Typical examples of an almost distance function are the distance function…

Study non-existence of complex ball quotients in Torelli locus.

In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the Meusnier spheres. We proved that if the curve is a principal line, the locus of the …

We study the singular locus of solutions to Hamilton-Jacobi equations with a Hamiltonian independent of $u$. In a previous paper, we proved that the singular locus is what we call a balanced split locus. In this paper, we find and classify all balanced split sets, identifying the cases where the only balanced split loc…

The paper describes the CR umbilical locus of a real ellipsoid in complex space.

New 2-spheres of revolution with simple cut locus structures.

Study on Blaschke locus with covariance metric properties.

Study conjugate locus in convex 3-manifolds using Jacobi fields.

Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.

The paper extends spacetime topology results using codimension 2 null cut locus properties.

New method calculates cut locus on surfaces without boundary.

We showed in another paper [arXiv:1103.1759] that every connected graph can be realized as the cut locus of some point on some riemannian surface $S$. Here, criteria for the orientability of $S$ are given, and are applied to classify the distinct, orientable, cut locus structures on graphs with four generating cycles.

We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus …

The intimate relationship between coherent states and geodesics is pointed out. For homogenous manifolds on which the exponential from the Lie algebra to the Lie group equals the geodesic exponential, and in particular for symmetric spaces, it is proved that the cut locus of the point $0$ is equal to the set of coheren…

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

Study on cut locus of submanifolds in Finsler geometry.

In the previous paper, the structure of the cut locus was determined for a class of surfaces of revolution homeomorphic to a cylinder. In this paper, we prove the structure theorem of the cut locus for a wider class of surfaces of revolution homeomorphic to a cylinder.

In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate poi…

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $π:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exceptio…

Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.

New method calculates cut locus on Riemannian manifolds using optimal transport.

The conjugate locus of a point $p$ in a surface $\mathcal{S}$ will have a certain number of cusps. As the point $p$ is moved in the surface the conjugate locus may spontaneously gain or lose cusps. In this paper we explain this `bifurcation' in terms of the vanishing of higher derivatives of the exponential map; we der…

New insights into surface group actions and entropy.

Study shows zero probability of cut locus for Fréchet mean on Riemannian manifolds.

In the paper we consider the Stiefel manifold $V_{n;k}$ as a principal $U(k)$- bundle over the Grassmann manifold and study the cut locus from the unit element. We gave the complete description of this cut locus on $V_{n;1}$ and presented the sufficient condition on the general case. At the end, we study the complement…

In the present paper we study the structure of the cut locus of a Randers rotational 2-sphere of revolution $(M, F = α+β)$. We show that in the case when the Gaussian curvature of the Randers surface is monotone along a meridian, the cut locus of a point $q\in M$ is a point on a subarc of the opposite half bending meri…

LOCUS separates brain network connectivity matrices efficiently.

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the …

Study on focal locus of submanifolds in Finsler manifolds, showing regularity and smoothness.

We study the small time asymptotics of the gradient and Hessian of the logarithm of the heat kernel at the cut locus, giving, in principle, complete expansions for both quantities. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic in…

We characterize the differentiable points of the distance function from a closed subset $N$ of an arbitrary dimensional Finsler manifold in terms of the number of $N$-segments. In the case of a 2-dimensional Finsler manifold, we prove the structure theorem of the cut locus of a closed subset $N$, namely that it is a lo…

Study geometry of surfaces glued along a curve.

The consequences for Berezin's quantization on symmetric spaces of the identity of the set of coherent vectors orthogonal to a fixed one with the cut locus are stated precisely. It is shown that functions expressing the coherent states, the covariant symbols of operators, the diastasis function, the characteristic and …

In this note, we prove the existence of a closed geodesic of positive length on any compact developable orbifold of dimension 3, 5, or 7. The argument uses the stratification of the singular locus, and reduces the problem of existence of a closed geodesic on a compact developable orbifold to the case of even dimensiona…

In this paper, we extend and complete the classification of the generic singularities of the 3D-contact sub-Riemmanian conjugate locus in a neighbourhood of the origin.

The paper proves CMC foliations for quasi-Fuchsian manifolds near the Fuchsian locus.

The focal locus $Σ_X$ of an affine variety $X$ is roughly speaking the (projective) closure of the set of points $O$ for which there is a smooth point $x \in X$ and a circle with centre $O$ passing through $x$ which osculates $X$ in $x$. Algebraic geometry interprets the focal locus as the branching locus of the endpoi…

We provide a very brief review of the description of colored invariants for the Hopf link in terms of characters, which need to be taken at a peculiar deformation of the topological locus, depending on one of the two representations associated with the two components of the link. Most important, we extend the descripti…

Locus scores predictions for risk, reducing large-loss events.

We proved in another paper that every connected graph can be realized as the cut locus of some point on some riemannian surface. Here we give upper bounds on the number of such realizations.

The paper connects cut locus, Thom space, and Morse-Bott functions in Riemannian geometry.

New foliations at infinity for quasi-Fuchsian manifolds near the Fuchsian locus are uniquely determined.

The study defines new surfaces with specific cut locus properties and provides conditions for their existence.