## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularities in Generic Geometry, S. Izumiya, G. Ishikawa, M. Yamamoto, K. Saji, T. Yamamoto and M. Takahashi, eds. (Tokyo: Mathematical Society of Japan, 2018), 345 - 363

### Constant mean curvature surfaces with $D_4$-singularities

Yuta Ogata and Keisuke Teramoto

#### Abstract

We study $D_4$-singularities of constant mean curvature (CMC) surfaces in Riemannian and semi-Riemannian spaceforms. We will give criteria for $D_4^{-}$-singularities, which are related to the Hopf differential. We also show the non-existence of (spacelike) CMC surfaces with $D_4^+$-singularities.

#### Article information

**Dates**

Received: 18 January 2016

Revised: 28 March 2017

First available in Project Euclid:
4 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1538618981

**Digital Object Identifier**

doi:10.2969/aspm/07810345

**Mathematical Reviews number (MathSciNet)**

MR3839953

**Zentralblatt MATH identifier**

07085111

**Subjects**

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]

Secondary: 53B30: Lorentz metrics, indefinite metrics 57R45: Singularities of differentiable mappings

**Keywords**

constant mean curvature surface parallel transformation singularity wave front

#### Citation

Ogata, Yuta; Teramoto, Keisuke. Constant mean curvature surfaces with $D_4$-singularities. Singularities in Generic Geometry, 345--363, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07810345. https://projecteuclid.org/euclid.aspm/1538618981