Sungwook Lee

A Weierstrass type representation for minimal surfaces in Sol

The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.

Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ31(−1)

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ℍ31(−1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in ℙSL2ℝ via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E31. The hyperbolic Gauß map of timelike surfaces in ℍ31(−1), which is a close analogue of the classical Gauß map is considered. It is discussed that the hyperbolic Gauß map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ℍ31(−1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gauß map and timelike surface of constant mean curvature ± 1 in ℍ31(−1) is studied.

Spacelike constant mean curvature 1 trinoids in de Sitter three-space

We derive a holomorphic spinor representation formula for s pacelike surfaces of constant mean curvature 1 in de Sitter 3-space, and use it to c onstruct examples of spacelike catenoids and trinoids with constant mean curvat re 1.

Flat Lorentz surfaces in anti-de Sitter 3-space and Gravitational Instantons

In this paper, we study flat Lorentz surfaces in anti-de Sitter 3-space ℍ13(−1) in terms of the second conformal structure. Those flat Lorentz surfaces can be represented in terms of a Lorentz holomorphic and a Lorentz anti-holomorphic data similarly to Weierstras representation formula. An analogue of hyperbolic Gaus map is considered for timelike surfaces in ℍ13(−1) and the relationship between the conformality (or the holomorphicity) of hyperbolic Gaus map and the flatness of a Lorentz surface is discussed. It is shown that flat Lorentz surfaces in ℍ13(−1) are associated with a hyperbolic Monge–Ampere equation. It is also known that Monge–Ampere equation may be regarded as a 2-dimensional reduction of the Einstein’s field equation. Using this connection, we construct a class of anti-self-dual gravitational instantons from flat Lorentz surfaces in ℍ13(−1).

Split‐quaternionic representation of SDYM SU(1,1) instantons in S−2 × S+2

Using split‐quaternions, we find explicit SDYM SU(1,1) instanton solutions in S−2 × S+2 which is the conformal compactification of the semi‐Euclidean 4‐spacetime R2+2 of split‐signature (−,−,+,+). It is also shown that SDYM and ASDYM fields in S−2 × S+2 can be described as simple split‐quaternionic 2‐forms.