Adam Szereszewski

L-isothermic and L-minimal surfaces

It has recently been shown that a second-order linear nonhomogeneous differential equation is associated with a surface with an isothermic representation of their lines of curvature (L-isothermic surface) (Schief et al 2007 J. Math. Phys. 48 073510). The 6-parameter group acting on linearly independent solutions of the homogeneous version of the latter equation generates a Laguerre transformation of the surface. The Weierstrass representation of the surfaces which are both L-isothermic and L-minimal is presented.

A Bäcklund transformation for L-isothermic surfaces

It is established that a Backlund transformation for L-isothermic surfaces is associated with a Darboux transformation for a non-homogeneous linear Schrodinger equation. A Lax pair for L-isothermic surfaces is presented and it is shown that a quartet of eigenfunctions contained therein may be explicitly represented in terms of linearly independent solutions of a linear Schrodinger equation with a potential involving the Backlund parameter. A permutability theorem is presented whereby L-isothermic surfaces may be constructed and the action of the Backlund transformation on a class of generalized Dupin cyclides is considered.

The Lamé equation in shell membrane theory

A classical nonlinear shell membrane system has recently been demonstrated to have underlying integrable structure. Here, a wide class of corresponding parallel membranes is shown to be generated via a Schrodinger equation of Lame type. This class is characterized by the existence of a multiplicity of stress distributions for a given membrane geometry. A variety of viable membrane geometries such as generalized Dupin cyclides are constructed explicitly together with the associated one-parameter families of stress resultants. A Lax pair which encapsulates both the Gauss-Mainardi-Codazzi and constrained equilibrium equations is also recorded.

Spacetimes foliated by nonexpanding and Killing horizons: Higher dimension

The theory of non-expanding horizons (NEH) geometry and the theory of near horizon geometries (NHG) are two mathematical relativity frameworks generalizing the black hole theory. From the point of view of the NEHs theory, a NHG is just a very special case of a spacetime containing an NEH of many extra symmetries. It can be obtained as the Horowitz limit of a neighborhood of an arbitrary extremal Killing horizon. An unexpected relation between the two of them, was discovered in the study of spacetimes foliated by a family of NEHs. The class of 4-dimensional NHG solutions (either vacuum or coupled to a Maxwell field) was found as a family of examples of spacetimes admitting a NEH foliation. In the current paper we systematically investigate geometries of the NEHs foliating a spacetime for arbitrary matter content and in arbitrary spacetime dimension. We find that each horizon belonging to the foliation satisfies a condition that may be interpreted as an invitation for a transversal extremal Killing horizon to exist. Assuming the existence of a transversal extremal Killing horizon, we derive all the spacetime metrics satisfying the vacuum Einsteins equations.

On shell membranes of Enneper type: generalized Dupin cyclides

It is demonstrated that a class of generalized Dupin cyclides arises naturally out of a classical system of equilibrium equations for shell membranes. This class consists of all families of parallel canal surfaces on which the lines of curvature are planar. Various examples of viable membrane geometries such as particular L-minimal surfaces are presented.

Near horizon geometries and black hole holograph

Two quasi local approaches to black holes are combined: Near Horizon Geometries (NHG) and stationary Black Hole Holographs (BHH). Necessary and sufficient conditions on BHH data for the emergence of NHGs as resulting vacuum solutions to Einsteins equations are found.

On Darboux's Approach to R-Separability of Variables ?

We discuss the problem of R-separability (separability of variables with a fac- tor R) in the stationary Schrodinger equation on n-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of R-separability in PDE (Laplace equation on E 3 ). According to Darboux R-separability amounts to two conditions: metric is isothermic (all its parametric surfaces are isothermic in the sense of both classical differential geometry and modern theory of solitons) and more- over when an isothermic metric is given their Lam e coefficients satisfy a single constraint which is either functional (when R is harmonic) or differential (in the opposite case). These two conditions are generalized to n-dimensional case. In particular we define n-dimensional isothermic metrics and distinguish an important subclass of isothermic metrics which we call binary metrics. The approach is illustrated by two standard examples and two less standard examples. In all cases the approach offers alternative and much simplified proofs or deriva- tions. We formulate a systematic procedure to isolate R-separable metrics. This procedure is implemented in the case of 3-dimensional Laplace equation. Finally we discuss the class of Dupin-cyclidic metrics which are non-regularly R-separable in the Laplace equation on E 3 .

The axial symmetry of Kerr without the rigidity theorem

Local condition that imply the no-hair property of black holes are completed. The conditions take the form of constraints on the geometry of the 2-dimensional crossover surface of black hole horizon. They imply also the axial symmetry without the rigidity theorem. This is the new result contained in this letter. The family of the solutions to our constraints is 2-dimensional and can be parametrized by the area and angular momentum. The constraints are induced by our assumption that the horizon is of the Petrov type D. Our result applies to all the bifurcated Killing horizons: inner/outer black hole horizons as well as cosmological horizons. Vacuum spacetimes with a given cosmological constant can be reconstructed from our solutions via Raczs black hole holograph.

Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

We present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Bäcklund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalization of the integrable discrete Tzitzéica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretization of affine spheres in affine differential geometry.

The Petrov type D equation on genus

Abstract The Petrov type D equation imposed on the 2-metric tensor and the rotation scalar of a cross-section of an isolated horizon can be used to uniquely distinguish the Kerr–(anti) de Sitter spacetime in the case the topology of the cross-section is that of a sphere. In the current paper we study that equation on closed 2-dimensional surfaces that have genus >0. We derive all the solutions assuming the embeddability in 4-dimensional spacetime that satisfies the vacuum Einstein equations with (possibly 0) cosmological constant. We prove all of them have constant Gauss curvature and zero rotation. Consequently, we provide a quasi-local argument for a black hole in 4-dimensional spacetime to have a topologically spherical cross-section.