International Journal of Mathematics

A three-dimensional Lorentzian hypersurface $x:M_1^3\to {ℝ}_1^4$ is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of ${ℝ}_1^4$. In this paper, using the projective light-cone model, we give a complete classification of those ones whose shape operators have two distinct real eigenvalues and cannot be diagonalizable. These hypersurfaces are conformal equivalent to cones, cylinders, or rotational hypersurfaces generated by B-scrolls (over null Frenet curves) in three-dimensional Lorentzian space forms.

We obtain new inversion formulas for the Radon transform and its dual between lines and hyperplanes in ${ℝ}^n$. The Radon transform in this setting is non-injective and the consideration is restricted to the so-called quasi-radial functions that are constant on symmetric clusters of lines. For the corresponding dual transform, which is injective, explicit inversion formulas are obtained both in the symmetric case and in full generality. The main tools are the Funk transform on the sphere, the Radon-John $d$-plane transform in ${ℝ}^n$, the Grassmannian modification of the Kelvin transform, and the Erdélyi–Kober fractional integrals.

There are two well-known parabolic split ${\mathrm{\text{G}}}_2$ geometries in dimension 5, $(2,3,5)$ distributions and ${\mathrm{\text{G}}}_2$ contact structures. Here we link these two geometries with yet another ${\mathrm{\text{G}}}_2$ related contact structure, which lives on a $7$-manifold. More precisely, we present a natural geometric construction that associates to a $(2,3,5)$ distribution a $7$-dimensional bundle endowed with a canonical Lie contact structure. We further study the relation between the canonical normal Cartan connections associated with the two structures and we show that the Cartan holonomy of the induced Lie contact structure reduces to ${\mathrm{\text{G}}}_2$. This motivates the study of the curved orbit decomposition associated with a ${\mathrm{\text{G}}}_2$ reduced Lie contact structure on a $7$-manifold. It is shown that, provided an additional curvature condition is satisfied, in a neighborhood of each point in the open curved orbit the structure descends to a $(2,3,5)$ distribution on a local leaf space. The closed orbit carries an induced ${\mathrm{\text{G}}}_2$ contact structure.

A compact Riemann surface $X$ of genus $g$ is called pseudo-real if it admits an anticonformal automorphism but no anticonformal involution. In this paper, we study pseudo-real $(q,n)$-gonal Riemann surfaces of genera greater or equal to two; these surfaces have anticonformal automorphisms of prime order $n$ such that the quotient spaces have genus $q$.

We give presentations of the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. As the volume conjecture states, the leading terms of the expansions present the hyperbolic volume and the Chern–Simons invariant of the complements of the knots. As coefficients of the expansions, we obtain a series of new invariants of the knots. This paper is a continuation of the previous papers [T. Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot, Quantum Topol. 7 (2016) 669–735; T. Ohtsuki and Y. Yokota, On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings, to appear in Math. Proc. Cambridge Philos. Soc.], where the asymptotic expansions of the Kashaev invariant are calculated for hyperbolic knots with five and six crossings. A technical difficulty of this paper is to use 4-variable saddle point method, whose concrete calculations are far more complicated than the previous papers.

Let $x:M\to {ℝ}^N$ be an $n$-dimensional compact self-shrinker in ${ℝ}^N$ with smooth boundary $\partial M$. In this paper, we study eigenvalues of the operator ${{ℒ}}_r$ on $M$, where ${{ℒ}}_r$ is defined by

We study the deformations of a smooth curve $C$ on a smooth projective $3$-fold $V$, assuming the presence of a smooth surface $S$ satisfying $C\subset S\subset V$. Generalizing a result of Mukai and Nasu, we give a new sufficient condition for a first order infinitesimal deformation of $C$ in $V$ to be primarily obstructed. In particular, when $V$ is Fano and $S$ is $K3$, we give a sufficient condition for $C$ to be (un)obstructed in $V$, in terms of $(-2)$-curves and elliptic curves on $S$. Applying this result, we prove that the Hilbert scheme ${Hilb}^{\mathrm{\text{sc}}}{V}_4$ of smooth connected curves on a smooth quartic $3$-fold $V_4\subset {\mathbb{P}}^4$ contains infinitely many generically non-reduced irreducible components, which are variations of Mumford’s example for ${Hilb}^{\mathrm{\text{sc}}}{\mathbb{P}}^3$.