My Old Publications in Mathematics

  • Maximal spacelike surfaces in a certain homogeneous Lorentzian 3-manifold. Preprint is available here and also available at arXiv:1503.06305.
  • Minimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold, Tohoku Mathematical Journal (2) 69 (2017), no. 4, 621-635. Preprint is available here and also available at arXiv:1503.02604.
  • Minimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold II, Differential Geometry - Dynamical Systems, Volume 18 (2016), 58-71. Preprint is available here.
  • Flat Lorentz Surfaces in Anti-de Sitter 3-Space and Gravitational Instantons, with Jun-ichi Inoguchi and Marianty Ionel, International Journal of Geometric Methods in Modern Physics (IJGMMP), 13 (2016), no. 2, 1650012, 19pp. Preprint is available here.
  • Timelike surfaces of revolution with constant mean curvature in de Sitter 3-space, with Jacob Martin, International Electronic Journal of Geometry, Volume 8, No. 1 (2015), 116-127. Preprint is available here.

    Jacob Martin was a promising high school (Oak Grove High School) senior when he joined me for the research project reported in this paper. The research has been done during Summer 2013-Fall 2013. Jacob is currently an undergraduate student majoring mathematics at Massachusetts Institute of Technology. The Department of Mathematics at the University of Southern Mississippi is committed to support nearby high schools by providing talented high school students opportunities for a research experience in mathematical sciences. This research was done as part of such an outreach effort.

    Animations: I made some animations in regard to this project.

    • Animation 1: Animation of CMC \(H=c\) timelike surfaces of revolution in \(\mathbb S^3_1(c^2)\) tending toward timelike catenoid in \(\mathbb R^{2+1}\) as \(c\to 0\).
    • Animation 2: Animation of minimal timelike surfaces of revolution in \(\mathbb S^3_1(c^2)\) tending toward timelike catenoid in \(\mathbb R^{2+1}\) as \(c\to 0\).
  • Spacelike surfaces of revolution with constant mean curvature in de Sitter 3-Space, Differential Geometry - Dynamical Systems, Volume 17 (2015), 81-96. Preprint is available here.

    Animations: I made some animations in regard to this project.

    • Animation 1: Animation of profiles curves for CMC \(H=c\) spacelike surfaces of revolution in de Sitter 3-space \(\mathbb S^3_1(c^2)\) tending toward the profile curve of spacelike catenoid in \(\mathbb R^{2+1}\) as \(c\to 0\).
    • Animation 2: Animation of CMC \(H=c\) spacelike surfaces of revolution in \(\mathbb S^3_1(c^2)\) tending toward spacelike catenoid in \(\mathbb R^{2+1}\) as \(c\to 0\).
    • Animation 3: Animation 2 with spacelike catenoid in \(\mathbb R^{2+1}\).
    • Animation 4: Animation of maximal spacelike surfaces of revolution in \(\mathbb S^3_1(c^2)\) tending toward spacelike catenoid in \(\mathbb R^{2+1}\) as \(c\to 0\).
  • Surfaces of Revolution with Constant Mean Curvature in Hyperbolic 3-Space, with Kinsey Zarske, Differential Geometry - Dynamical Systems, Volume 16 (2014), 203-218. Preprint is available here.

    Kinsey Zarske is an undergraduate student majoring mathematics and physics at the University of Southern Mississippi. A part of research presented in this paper was done as her undergradaute research project under my direction.

    Animations: I made some animations in regard to this project.

    • Animation 1: Animation of profiles curves for CMC \(H=c\) surfaces of revolution in \(\mathbb H^3(-c^2)\) tending toward the profile curve of catenoid in \(\mathbb E^3\) as \(c\to 0\).
    • Animation 2: Animation of CMC \(H=c\) surfaces of revolution in \(\mathbb H^3(-c^2)\) tending toward catenoid in \(\mathbb E^3\) as \(c\to 0\).
    • Animation 3: Animation 2 with catenoid in \(\mathbb E^3\).
    • Animation 4: Animation of minimal surfaces of revolution in \(\mathbb H^3(-c^2)\) tending toward catenoid in \(\mathbb E^3\) as \(c\to 0\).

    • Errata: The authors made some minor errors in the paper and unfortunately they were discovered after the paper was published. The following are the corrections of these errors.

    • Erratum 1. In the paper, the authors mentioned that the Lawson correspondence implies that there is no surface in hyperbolic 3-space \(\mathbb{H}^3(-c^2)\) with \(H=0\) unless \(c=0\) in which case the space is Euclidean 3-space \(\mathbb{E}^3\) (p.208). This is incorrect. The Lawson correspondence only implies that there are no constant mean curvature surfaces in \(\mathbb{E}^3\) that are corresponded to \(H=0\) surfaces in \(\mathbb{H}^3(-c^2)\). However, it can be easily shown that there are no conformal surfaces of revolution with \(H=0\) in \(\mathbb{H}^3(-c^2)\):

      It follows from equation (5.4) on p.209 that \(H=0\) if and only if \(-h''(u)+h(u)=0\). Differentiating the conformality condition (5.3) on p.209, we obtain\[h'(u)(-h''(u)+h(u))=ce^{2cu}.\] Hence, we see that if \(H=0\) then \(c=0\).

    • Erratum 2. The equation (A.5) on p.213 should be read: \begin{align*} H&=\frac{1}{2}\mathrm{tr} S=\frac{a+d}{2}\\ &=\frac{1}{2}\frac{\langle S(\varphi_u),\varphi_u\rangle\langle\varphi_v,\varphi_v\rangle+\langle S(\varphi_v),\varphi_v\rangle\langle\varphi_u,\varphi_u\rangle-2\langle S(\varphi_u),\varphi_v\rangle\langle\varphi_u,\varphi_v\rangle}{\langle\varphi_u,\varphi_u\rangle\langle\varphi_v,\varphi_v\rangle-\langle\varphi_u,\varphi_v\rangle^2}. \end{align*}
  • Lightlike surfaces in Minkowski 3-space, with J. Inoguchi, International Journal of Geometric Methods in Modern Physics (IJGMMP) 6 (2009), Issue 2, 267-283. DOI No: 10.1142/S0219887809003552.
  • Null curves in Minkowski 3-space., with J. Inoguchi, International Electronic Journal of Geometry 1 (2008), No. 2, 40-83.
  • Maximal surfaces in a certain 3-dimensional homogeneous spacetime, Differential Geometry and Its Applications 26 (2008), Issue 5, 536-543.
  • A Weierstrass type representation for minimal surfaces in Sol, with J. Inoguchi, Proc. Amer. Math. Soc. 136 (2008), 2209-2216. Preprint is also available at arXiv:math/0609722.
  • Weierstrass representation for timelike minimal surfaces in Minkowski 3-space, Proceedings of the Midwest Geometry Conference held at the University of Oklahoma, May 5-7, 2006, Communications in Mathematical Analysis, Conf. 01(2008), 11-19. Preprint is available at arXiv:math/0608726.
  • Timelike constant mean curvature surfaces of revolution in Minkowski 3-space, with J. H Varnado, Differential Geometry and Dynamical Systems 9, No. 1 (2007), 82-102.

    • This is a paper from an undergraduate research project.

  • Spacelike constant mean curvature surfaces of revolution in Minkowski 3-space, with J. H Varnado, Differential Geometry and Dynamical Systems 8, no. 1 (2006), 144-165.

    • This is a paper from an undergraduate research project.

  • Spacelike constant mean curvature 1 trinoids with singularities in de Sitter 3-space, with S.-D. Yang, Osaka Journal of Mathematics 43 (2006), 641-663.
  • Timelike surfaces of constant mean curvature one in anti-de Sitter 3-space, Annals of Global Analysis and Geometry 29, no. 4 (June, 2006), 355-401. DOI: 10.1007/s10455-006-9030-z. Preprint is available at arXiv:math/0607104.
  • Spacelike surfaces of constant mean curvature one in de Sitter 3-space, Illinois Journal of Mathematics 49, Issue 1 (Spring 2005), 63-98.
  • Spacelike CMC 1 surfaces in de Sitter 3-space: their construction and some examples, Differential Geometry and Dynamical Systems 7 (2005), 49-73.