GENUS MINIMAL k-NOIDS AND SADDLE TOWERS IN

For each $k\geq 3$ , we construct a $1$ -parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb {H}^2\times \mathbb {R}$ with genus $1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi $ . Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.

Triality and the consistent reductions on AdS3 × S3

We study compactifications on AdS3×S3 and deformations thereof. We exploit the triality symmetry of the underlying duality group SO(4,4) of three-dimensional supergravity in order to construct and relate new consistent truncations. For non-chiral D = 6, $$ \mathcal{N} $$ N 6d = (1, 1) supergravity, we find two different consistent truncations to three-dimensional supergravity, retaining different subsets of Kaluza-Klein modes, thereby offering access to different subsectors of the full nonlinear dynamics. As an application, we construct a two-parameter family of AdS3 × M3 backgrounds on squashed spheres preserving U(1)2 isometries. For generic value of the parameters, these solutions break all supersymmetries, yet they remain perturbatively stable within a non-vanishing region in parameter space. They also contain a one-parameter family of $$ \mathcal{N} $$ N = (0, 4) supersymmetric AdS3 × M3 backgrounds on squashed spheres with U(2) isometries. Using techniques from exceptional field theory, we determine the full Kaluza-Klein spectrum around these backgrounds.

Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54

We study the out-of-equilibrium dynamics of the quantum cellular automaton Rule 54 using a time-channel approach. We exhibit a family of (non-equilibrium) product states for which we are able to describe exactly the full relaxation dynamics. We use this to prove that finite subsystems relax to a one-parameter family of Gibbs states. We also consider inhomogeneous quenches. Specifically, we show that when the two halves of the system are prepared in two different solvable states, finite subsystems at finite distance from the centre eventually relax to the non-equilibrium steady state (NESS) predicted by generalised hydrodynamics. To the best of our knowledge, this is the first exact description of the relaxation to a NESS in an interacting system and, therefore, the first independent confirmation of generalised hydrodynamics for an inhomogeneous quench.

Ribaucour surfaces of harmonic type

We introduce the class of Ribaucour surfaces of harmonic type (in short HR-surfaces) that generalizes the Ribaucour surfaces related to a problem posed by Élie Cartan. We obtain a Weierstrass-type representation for these surfaces which depends on three holomorphic functions. As application, we classify the HR-surfaces of rotation, present examples of complete HR-surfaces of rotation with at most two isolated singularities and an example of a complete HR-surface of rotation with one catenoid type end and one planar end. Also, we present a 5-parameter family of cyclic HR-surfaces foliated by circles in non-parallel planes. Moreover, we classify the isothermic HR-surfaces with planar lines of curvature.

Maximisers for Strichartz inequalities on the torus

We study the existence of maximisers for a one-parameter family of Strichartz inequalities on the torus. In general, maximising sequences can fail to be precompact in L 2 ( T ) , and maximisers can fail to exist. We provide a sufficient condition for precompactness of maximising sequences (after translation in Fourier space), and verify the existence of maximisers for a range of values of the parameter. Maximisers for the Strichartz inequalities correspond to stable, periodic (in space and time) solutions of a model equation for optical pulses in a dispersion-managed fiber.

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector fields possessing a polynomial integral of degree 6 whose level curves are of genus 1, as well. These vector fields are non-homogeneous generalizations of reduced Nahm systems for magnetic monopoles with icosahedral symmetry, introduced by Hitchin, Manton and Murray. The straightforward Kahan discretization of these novel non-homogeneous systems is non-integrable. However, this drawback is repaired by introducing adjustments of order $$O(\epsilon ^2)$$ O ( ϵ 2 ) in the coefficients of the discretization, where $$\epsilon $$ ϵ is the stepsize.

Q-balls meet fuzzballs: non-BPS microstate geometries

We construct a three-parameter family of non-extremal microstate geometries, or “microstrata”, that are dual to states and deformations of the D1-D5 CFT. These families are non-extremal analogues of superstrata. We find these microstrata by using a Q-ball-inspired Ansatz that reduces the equations of motion to solving for eleven functions of one variable. We then solve this system both perturbatively and numerically and the results match extremely well. We find that the solutions have normal mode frequencies that depend upon the amplitudes of the excitations. We also show that, at higher order in perturbations, some of the solutions, having started with normalizable modes, develop a “non-normalizable” part, suggesting that the microstrata represent states in a perturbed form of the D1-D5 CFT. This paper is intended as a “Proof of Concept” for the Q-ball-inspired approach, and we will describe how it opens the way to many interesting follow-up calculations both in supergravity and in the dual holographic field theory.