Miyuki Koiso | Last modified date：2018.03.23 |

Graduate School

Undergraduate School

Homepage

##### http://www.imi.kyushu-u.ac.jp/eng/academic_staffs/view/85

Academic Degree

Doctor of Science

Country of degree conferring institution (Overseas)

No

Field of Specialization

mathematics

Total Priod of education and research career in the foreign country

04years00months

Research

**Research Interests**

- Global analysis on geometric variational problems and its applications

keyword : mathematics, geometry, differential geometry, variational problem, global analysis

2008.10～2017.03.

**Current and Past Project**

- Joint research with Prof. Jaigyoung Choe (Korean Institute for Advanced Study, Korea) on a free boundary problem for surfaces with constant mean curvature.
- Joint research on geometric variational problems for hypersurfaces; Miyuki Koiso (Kyushu U., Japan) and Bennett Palmer (Idaho State U., USA)
- Joint research on stability and bifurcation for surfaces with constant mean curvature and their generalizations; Miyuki Koiso (Kyushu U, Japan), Bennett Palmer (Idaho State U., USA), and Paolo Piccione (Univ. of Sao Paulo, Brazil)

**Academic Activities**

**Papers**

**Presentations**

1. | 小磯 深幸, Local structure of the space of all triply periodic minimal surfaces in R^3, Workshop "Geometric aspects on capillary problems and related topics", 2015.12, [URL], We study the space of triply periodic minimal surfaces in ${\mathds R}^3$, giving a result on the local rigidity and a result on the existence of bifurcation. We prove that, near a triply periodic minimal surface with nullity three, the space of triply periodic minimal surfaces consist of a smooth five-parameter family of pairwise non-homothetic surfaces. On the other hand, if there is a smooth one-parameter family of triply periodic minimal surfaces $\{X_t\}_t$ containing $X_0$ where the Morse index jumps by an odd integer, it will be proved the existence of a bifurcating branch issuing from $X_0$. We also apply these results to several known examples.. |

2. | 小磯 深幸, Stable capillary hypersurfaces in a wedge and uniqueness of the minimizer, Asymptotic Problems: Elliptic and Parabolic Issues, 2015.06, [URL], Let $\Sigma$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes $\Pi_1$, $\Pi_2$ in $\mathbb R^{n+1}$. Suppose $\Sigma$ meets each $\Pi_i$ in constant contact angle not less than $\pi/2$. We prove that if $\partial \Sigma$ is embedded for $n=2$, or if $\partial\Sigma$ is convex for $n\geq3$, then $\Sigma$ is part of the round sphere.. |

3. | 小磯 深幸, Bifurcation theory for minimal and constant mean curvature surfaces, Conference on Geometry, 2014.03, [URL], We construct general criteria for existence and nonexistence of ( continuous and discrete) bifurcation for minimal and constant mean curvature surfaces. For continuous bifurcation, we also give a criterion for stability for each surface in the bifurcation branch. We apply our general results to several concrete boundary value problems. Especially, we mention the existence of unknown examples of triply periodic minimal surfaces in the Euclidean three-space which are close to known examples. This talk is based on joint work with Bennett Palmer (Idaho State U., USA) and Paolo Piccione (University of Sao Paulo, Brazil), and joint work with Paolo Piccione and Toshihiro Shoda (Saga U., Japan). . |

4. | 小磯 深幸, Stable capillary hypersurfaces in a wedge and uniqueness of the minimizer, The second Japanese-Spanish workshop on Differential Geometry, 2014.02, [URL], We study a variational problem for immersed hypersurfaces in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. The total energy of each hypersurface is the $n$-dimensional surface area and a positive ``wetting energy'' on the supporting hyperplanes, and we impose the $(n+1)$-dimensional volume constraint enclosed by the hypersurfaces. Any stationary hypersurface $\Sigma$ is a hypersurface with constant mean curvature which meets each supporting hyperplane with constant contact angle, and it is said to be stable if the second variation of the energy is nonnegative for all admissible variations. We show that if $\Sigma$ is stable and is disjoint from the edge of the wedge, and if $\partial \Sigma$ is embedded for $n=2$, or if $\partial\Sigma$ is convex for $n\geq3$, then $\Sigma$ is part of the hypersphere. Our results also show that the space of stable solutions is not continuous with respect to the variation of the boundary condition. Moreover, we mention the uniqueness of the minimizer. This is joint work with Jaigyoung Choe (KIAS, Korea).. |

5. | 小磯 深幸, Geometry of hypersurfaces with constant anisotropic mean curvature, The 2013 Annual Meeting of the Taiwan Mathematical Society, 2013.12, [URL], A surface with constant anisotropic mean curvature (CAMC surface) is a stationary surface of a given anisotropic surface energy functional for volume-preserving variations. For example, minimal surfaces and surfaces with constant mean curvature in the Euclidean space and those in the Lorentz-Minkowski space are regarded as CAMC surfaces for a certain special anisotropic surface energy. The minimizer of an anisotropic surface energy among all closed surfaces enclosing the same volume is called the Wulff shape, and the minimizer among surfaces with free boundary on a given support surface is sometimes called the Winterbottom shape. These concepts can be naturally generalized to higher dimensions, and they have many applications inside and outside mathematics. In this talk, we give fundamental geometric properties of CAMC hypersurfaces and recent progress in the research on the stability of CAMC hypersurfaces with free or fixed boundaries.. |

6. | 小磯 深幸, Free boundary problem for surfaces with constant mean curvature, International Workshop on Special Geometry and Minimal Submanifolds, 2013.08, [URL], We study embedded surfaces of constant mean curvature with free boundary in given supporting planes in the euclidean three-space. We assume that each considered surface meets the supporting planes with constant contact angle. These surfaces are characterized as equilibrium surfaces of the variational problem of which the total energy is the surface area and a wetting energy (that is a weighted area of the domains in the supporting planes bounded by the boundary of the considered surface) with volume constraint. An equilibrium surface is said to be stable if the second variation of the energy is nonnegative for all volume-preserving variations satisfying the boundary condition. We are interested in determining all (stable) solutions. At present in literature, only for some special cases, for example, the supporting planes are either just a single plane or two parallel planes and the wetting energy is nonnegative, all stable solutions are known. We discuss recent progress of this subject and show the space of solutions is not continuous with respect to the boundary condition. . |

7. | 小磯 深幸, Bernstein-type theorems for surfaces with constant anisotropic mean curvature and CMC surfaces in the Lorentz-Minkowski space, 7th International Meeting on Lorentzian Geometry, 2013.07, [URL], A surface with constant anisotropic mean curvature (CAMC surface) is a stationary surface of a given anisotropic surface energy functional for volume-preserving variations. Surfaces with constant mean curvature (CMC surfaces) in the Lorentz-Minkowski space are regarded as CAMC surfaces for a certain special anisotropic surface energy. In this talk, we show that if a complete CAMC surface for a uniformly convex anisotropic surface energy in the euclidean three-space is a graph of a function in a whole plane, then it is a plane. Moreover, by using a similar method, we show that if a spacelike complete CMC surface in the Lorentz- Minkowski three-space satisfies a certain condition on the order of divergence of its Gauss map, then it is a plane.. |

8. | 小磯 深幸, Non-convex anisotropic surface energy and zero mean curvature surfaces in the Lorentz-Minkowski space, The 5th OCAMI-TIMS Joint International Workshop on Differential Geometry and Geometric Analysis, 2013.03, [URL], We study stationary surfaces of anisotropic surface energies in the euclidean three-space which are called anisotropic minimal surfaces. Usual minimal surfaces, zero mean curvature spacelike surfaces and timelike surfaces in the Lorenz-Minkowski space are regarded as anisotropic minimal surfaces for certain special axisymmetric anisotropic surface energies. In this talk, for any axisymmetric anisotropic surface energy, we show that, a surface is both a minimal surface and an anisotropic minimal surface if and only if it is a right helicoid. We also construct new examples of anisotropic cyclic minimal surfaces for certain reasonable classes of energy density. Our examples include zero mean curvature timelike surfaces and spacelike surfaces of catenoid-type and Riemann- type. This is a joint work with Atsufumi Honda (Tokyo Institute of Technology). . |

9. | 小磯 深幸, Geometry of isoperimetric-type problems modeled on interfaces on micrometre scale, Workshop on Geometry of Interfaces and Capillarity, 2012.06, [URL], We study geometry of isoperimetric-type problems modeled on interfaces on micrometre scale among two or three different phases. Our main subject is surfaces with constant (anisotropic) mean curvature with free or fixed boundary. We discuss existence, stability, bifurcation, and topological transition for solutions.. |

10. | Miyuki Koiso, Geometric analysis for variational problems of isoperimetric type, Invited Organized Talk, Annual meeting of the Mathematical Society of Japan, Tokyo University of Science, March 26, 2012., [URL]. |

11. | Miyuki Koiso， Bifurcation and stability for solutions of isoperimetric problems, Isoperimetric problems, space-filling, and soap bubble geometry, Mar 19, 2012 - Mar 23, 2012, ICMS (International Center for Mathematical Sciences), 15 South College Street Edinburgh, UK, Organisers: Cox, Simon (Institute of Mathematics and Physics), Morgan, Frank (Williams College), Sullivan, John (Technische Universitat Berlin). , [URL]. |

12. | Miyuki Koiso， Pitchfork bifurcation for hypersurfaces with constant mean curvature, The 10th Pacific Rim Geometry Conference 2011 Osaka-Fukuoka (December 1-5, Osaka City University, December 7-9, Kyushu University), December 7, 2011., [URL]. |

13. | Miyuki Koiso， Stability of surfaces with constant anisotropic mean curvature and applications to physical phenomena, III Encontro Paulista de Geometria (San Paulo, Brazil), August 9, 2011 , [URL]. |

14. | Geometric variational problems and bifurcation theory, [URL]. |

15. | Geometry of hypersurfaces with constant anisotropic mean curvature, [URL]. |

16. | Stability and bifurcation for surfaces with constant mean curvature and their generalizations, [URL]. |

17. | Stability and bifurcation for surfaces with constant mean curvature and their generalizations, [URL]. |

18. | Stability and bifurcation for solutions of isoperimetric type problems, [URL]. |

19. | , [URL]. |

20. | , [URL]. |

21. | , [URL]. |

**Membership in Academic Society**

- Society for Science on Form, Japan
- Japan Society of Mathematical Education
- Mathematical Society of Japan

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