小磯 深幸（こいそ みゆき） | データ更新日：2022.01.22 |

教授 ／
マス・フォア・インダストリ研究所
基礎理論研究部門

**原著論文**

1. | Miyuki Koiso, Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-similar solutions of anisotropic mean curvature flow, Advanced Studies in Pure Mathematics, T. Hoffmann, M. Kilian, K. Leschke, F. Martin (Eds.), Minimal surfaces: integrable systems and visualisation, Springer Proceedings in Mathematics & Statistics 349 (scheduled to be published in June, 2021)., 2021.06, [URL], An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered surface, and it is a generalization of surface area. Equilibrium surfaces with volume constraint are called CAMC (constant anisotropic mean curvature) surfaces and they are not smooth in general. We show that, if the energy density function is two times continuously differentiable and convex, then, like isotropic (constant mean curvature) case, the uniqueness for closed stable CAMC surfaces holds under the assumption of the integrability of the anisotropic principal curvatures. Moreover, we show that, unlike the isotropic case, uniqueness of closed embedded CAMC surfaces with genus zero in the three-dimensional euclidean space does not hold in general. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. These results are generalized to hypersurfaces in the Euclidean space with g. |

2. | Yuta Hatakeyama, Miyuki Koiso, Stability of helicoidal surfaces with constant mean curvature, International Journal of Mathematics for Industry, 10.1142/S2661335220500070, 12, 1, 2050007, 12-1, 2050007 (23 pages), 2021.02, Motivated by an experimental result on the shape of liquid water supported by a small stainless helical wire, we find a class of stable helicoidal convex surfaces with constant mean curvature whose boundary consists of a single helix and two short arcs.. |

3. | Miyuki Koiso, Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature, Advanced Studies in Pure Mathematics, 10.2969/aspm/08510239, 85, 239-258, 2020.12, [URL], We study a variational problem for piecewise-smooth hypersurfaces in the (n+1)-dimensional Euclidean space. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface, which is a generalization of the area of surfaces. The minimizer of such an energy among all closed hypersurfaces enclosing the same (n+1)-dimensional volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium hypersurfaces of this energy for volume-preserving variations are not smooth in general. In this article we give recent results on the uniqueness and non-uniqueness for closed equilibria. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. This article is an announcement of some forthcoming papers.. |

4. | Atsufumi Honda, Yu Kawakami, Miyuki Koiso, Syunsuke Tori, Heinz-type mean curvature estimates in Lorentz-Minkowski space, Revista Matematica Complutense, https://doi.org/10.1007/s13163-020-00373-9, 2020.10, We provide a unified description of Heinz-type mean curvature estimates under an assumption on the gradient bound for space-like graphs and time-like graphs in the Lorentz-Minkowski space. As a corollary, we give a unified vanishing theorem of mean curvature for these entire graphs of constant mean curvature.. |

5. | Miyuki Koiso, Uniqueness of closed equilibrium hypersurfaces for anisotropic surface energy and application to a capillary problem, Mathematical and Computational Applications, https://doi.org/10.3390/mca24040088, 24, 2019.10, We study a variational problem for hypersurfaces in the Euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density that depends on the surface normal over the considered hypersurface, which was introduced to model the surface tension of a small crystal. The purpose of this paper is two-fold. First, we give uniqueness and nonuniqueness results for closed equilibria under weaker assumptions on the regularity of both considered hypersurfaces and the anisotropic surface energy density than previous works and apply the results to the anisotropic mean curvature flow. This part is an announcement of two forthcoming papers by the author. Second, we give a new uniqueness result for stable anisotropic capillary surfaces in a wedge in the three-dimensional Euclidean space.. |

6. | Miyuki Koiso, Geometry of anisotropic surface energy, RIMS Kokyuroku, 2068, 57-68, 2018.04. |

7. | Miyuki Koiso, Paolo Piccione, Toshihiro Shoda, On bifurcation and local rigidity of triply periodic minimal surfaces in R^3, Annales De L'Institut Fourier, https://doi.org/10.5802/aif.3222, 68, 6, 2743-2778, 2018.11, We study the space of triply periodic minimal surfaces in 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 consists 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 is proved that there exists a bifurcating branch issuing from X_0. We also apply these results to several known examples.. |

8. | Miyuki Koiso, Atsufumi Honda, Kentaro Saji, Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space, Hokkaido Mathematical Journal, ４７, 2, 2018.06, Fold singular points play important roles in the theory of maximal surfaces. For example, if a maximal surface admits fold singular points, it can be extended to a timelike minimal surface analytically. Moreover, there is a duality between conelike singular points and folds. In this paper, we investigate fold singular points on spacelike surfaces with non-zero constant mean curvature (spacelike CMC surfaces). We prove that spacelike CMC surfaces do not admit fold singular points. Moreover, we show that the singular point set of any conjugate CMC surface of a spacelike Delaunay surface with conelike singular points consists of (2,5)-cuspidal edges.. |

9. | Miyuki Koiso, Bennett Palmer, Paolo Piccione, Stability and bifurcation for surfaces with constant mean curvature, Journal of the Mathematical Society of Japan, 69, 4, 1519-1554, 2017.10, We give criteria for the existence of smooth bifurcation branches of fixed boundary CMC surfaces in R^3, and we discuss stability/instability issues for the surfaces in bifurcating branches. To illustrate the theory, we discuss an explicit example obtained from a bifurcating branch of fixed boundary unduloids in R^3.. |

10. | Miyuki Koiso, Bennett Palmer, Higher order variations of constant mean curvature surfaces, Calculus of Variations and PDE's, 10.1007/s00526-017-1246-1, 2017.10, We study the third and fourth variation of area for a compact domain in a constant mean curvature surface when there is a Killing field on R^3 whose normal component vanishes on the boundary. Examples are given to show that, in the presence of a zero eigenvalue, the non negativity of the second variation has no implications for the local area minimization of the surface.. |

11. | Miyuki Koiso, A. Honda, M. Kokubu, M. Umehara, K. Yamada, Mixed type surfaces with bounded mean curvatures in 3-dimensional space-times, Differential Geometry and its Applications, https://doi.org/10.1016/j.difgeo.2017.03.009, 52, 64-77, 2017.03, In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvature and investigate their properties.. |

12. | Miyuki Koiso, Jaigyoung Choe, Stable capillary hypersurfaces in a wedge, Pacific Journal of Mathematics, 10.2140/pjm.2016.280.1, 280, 1, 1-15, 2015.12, Let $\Sigma$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $\Sigma$ meets those two hyperplanes in constant contact angles $\ge \pi/2$ and is disjoint from the edge of the wedge, and suppose that $\partial\Sigma$ consists of two smooth components with one in each hyperplane of the wedge. It is proved that if $\partial \Sigma$ is embedded for $n=2$, or if each component of $\partial\Sigma$ is convex for $n\geq3$, then $\Sigma$ is part of the sphere. And the same is true for $\Sigma$ in the half-space of $\mathbb R^{n+1}$ with connected boundary $\partial\Sigma$.. |

13. | Miyuki Koiso, Bennett Palmer, Paolo Piccione, Bifurcation and symmetry breaking of nodoids with fixed boundary, Advances in Calculus of Variations, 10.1515/acv-2014-0011, 8, 4, 337-370, published on line on June 18, 2014, 2015.10, We prove bifurcation results for (compact portions of) nodoids in R^3, whose boundary consists of two fixed coaxial circles of the same radius lying in parallel planes. Degeneracy occurs at an infinite discrete sequence of instants, that are divided into four classes. Different types of bifurcation and break of symmetry occur at each instant of three of the four classes; bifurcation does not occur at the degeneracy instants of the fourth class.. |

14. | Miyuki Koiso, Bennett Palmer, Stable surfaces with constant anisotropic mean curvature and circular boundary, Proceedings of the American Mathematical Society, DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11892-7, 141, 11, 3817-3823, 2013.11, [URL], We show that, for an axially symmetric anisotropic surface energy, only stable disc-type surfaces with constant anisotropic mean curvature bounded by a circle which lies in a plane orthogonal to the rotation axis of the Wulff shape are rescalings of parts of the Wulff shape and the flat disc.. |

15. | 小磯 深幸, Stability of hypersurfaces with constant mean curvature and applications to isoperimetric problems, RIMS Kokyuroku, 1850, 23-33, 2013.09. |

16. | 小磯 深幸, Non-convex anisotropic surface energy and zero mean curvature surfaces in the Lorentz-Minkowski space, Journal of Math-for-Industry, 5, 73-82, 2013.04, [URL], An anisotropic surface energy functional is the integral of an energy density function over a surface. The energy density depends on the surface normal at each point. The usual area functional is a special case of such a functional. We study stationary surfaces of anisotropic surface energies in the euclidean three-space which are called anisotropic minimal surfaces. 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 minimal surfaces, which include zero mean curvature surfaces in the three-dimensional Lorentz-Minkowski space as special cases. . |

17. | Miyuki Koiso and Bennett Palmer, Equilibria for anisotropic surface energies with wetting and line tension, Calculus of Variations and Partial Differential Equations, 43, 3, 555-587, 2012.01, We study the stability of surfaces trapped between two parallel planes with free boundary on these planes. The energy functional consists of anisotropic surface energy, wetting energy, and line tension. Equilibrium surfaces are surfaces with constant anisotropic mean curvature. We study the case where the Wulff shape is of ``product form'', that is, its horizontal sections are all homothetic and has a certain symmetry. Such an anisotropic surface energy is a natural generalization of the area of the surface. Especially, we study the stability of parts of anisotropic Delaunay surfaces which arise as equilibrium surfaces. They are surfaces of the same product form of the Wulff shape. We show that, for these surfaces, the stability analysis can be reduced to the case where the surface is axially symmetric and the functional is replaced by an appropriate axially symmetric one. Moreover, we obtain necessary and sufficient conditions for the stability of anisotropic sessile drops.. |

18. | 小磯 深幸, Stability of non liquid bridges, Mathematica Contemproanea, 40, 243-260, 2011.08, We give a new, simpler proof characterizing the stable non liquid bridges. Numerical examples are given which show that the conditions imposed on the functionals in these theorems are essential. We also show that arbitrary convex non liquid bridges are always stable.. |

19. | Miyuki Koiso and Bennett Palmer, Anisotropic umbilic points and Hopf's theorem for surfaces with constant anisotropic mean curvature, Indiana University Mathematics Journal, 59, 1, 79-90, 2010.05, 非等方的表面エネルギーは、曲面の各点における法線方向に依存するエネルギー密度の曲面上での総和 (積分) である。与えられたエネルギー密度関数に対し、同じ体積を囲む閉曲面の中での非等方的表面エネルギーの最小解は（平行移動を除き）一意的に存在し、Wulff図形と呼ばれている。より一般に、囲む体積を変えない変分に対する非等方的表面エネルギーの臨界点は、非等方的平均曲率一定曲面となる。本論文では、3次元ユークリッド空間において、Wulff図形が滑らかな狭義凸曲面であるという仮定のもとで、種数0の非等方的平均曲率一定閉曲面は平行移動と相似を除きWulff図形に限ることを証明した。. |

20. | Miyuki Koiso and Bennett Palmer, Anisotropic Surface Energy, Proceedings of the 16th OCU International Academic Symposium 2008 "Riemann Surfaces, Harmonic Maps and Visualization", OCAMI Studies Volume 3, Osaka Municipal University Press, 105-117, 2010.03. |

21. | Miyuki Koiso and Bennett Palmer, Equilibria for anisotropic surface energies and the Gielis formula, Forma, 23, 1, 1-8, 2008.12. |

22. | Miyuki Koiso and Bennett Palmer, Rolling construction for anisotropic Delaunay surfaces, Pacific Journal of Mathematics, 234, 2, 345-378, Vol.234, No.2, pp.345-378, 2008.02. |

23. | Miyuki Koiso and Bennett Palmer, Uniqueness theorems for stable anisotropic capillary surfaces, SIAM Journal on Mathematical Analysis , 39, 3, 721-741, Vol.39, No.3, pp.721-741, 2007.08. |

24. | Miyuki Koiso and Bennett Palmer, Anisotropic capillary surfaces with wetting energy, Calculus of Variations and Partial Differential Equations, 29, 3, 295-345, Vol.29, No.3, pp.295-345, 2007.07. |

25. | Miyuki Koiso and Bennett Palmer, Stability of anisotropic capillary surfaces between two parallel planes, Calculus of Variations and Partial Differential Equations, 25, 3, 275-298, Vol.25, No.3, pp.275-298, 2006.03. |

26. | Miyuki Koiso and Bennett Palmer, Geometry and stability of surfaces with constant anisotropic mean curvature, Indiana University Mathematics Journal, 54, 6, 1817-1852, Vol.54, No.6, pp.1817-1852, 2005.12. |

27. | Miyuki Koiso and Bennett Palmer, On a variational problem for soap films with gravity and partially free boundary, Journal of the Mathematical Society of Japan, 57, 2, 333-355, Vol.57, No.2, pp.333-355, 2005.04. |

28. | Miyuki Koiso and Bennett Palmer, Geometry and stability of bubbles with gravity, Indiana University Mathematics Journal, 54, 1, 65-98, Vol.54, No.1, pp.65-98, 2005.01. |

29. | Miyuki Koiso, 非等方的平均曲率一定曲面の幾何学的性質と安定性, 京都大学数理解析研究所講究録, 1403, 1-9, Vol.1403, pp.1-9, 2004.11. |

30. | Miyuki Koiso, Stability for critical points of a one-dimensional geometric variational problem with constraint, 京都大学数理解析研究所講究録, 1292, 146-161, Vol.1292, pp.146-161, 2002.10. |

31. | Miyuki Koiso, Stability of surfaces with constant mean curvature in three-dimensional space forms, "Differential Geometry, Valencia 2001" (Ed. by O. Gil-Medrano and V. Miquel), World Scientific Publishing, River Edge, NJ., 187-196, pp.187-196, 2002.07. |

32. | Miyuki Koiso, Deformation and stability of surfaces with constant mean curvature, Tohoku Mathematical Journal (2, 54, 1, 145-159, Vol.54, No.1, pp.145-159, 2002.03. |

33. | Miyuki Koiso, A note on the stability of minimal surfaces with branch points, Tohoku Mathematical Publications, 20, 99-106, Vol.20, pp.99-106, 2001.09. |

34. | Miyuki Koiso, Delaunay 曲面について, Bulletin of Kyoto University of Education, Ser.B, 97, 13-33, Vol.97, PP.13-33, 2000.09. |

35. | Miyuki Koiso, The uniqueness for stable surfaces of constant mean curvature with free boundary on a plane, Bulletin of Kyoto University of Education, Ser.B, 97, 1-12, Vol.97, pp.1-12, 2000.09. |

36. | Miyuki Koiso, The uniqueness for stable surfaces of constant mean curvature with free boundary, Bulletin of Kyoto University of Education, Ser.B, 94, 1-7, Vol.94, PP.1-7, 1999.03. |

37. | Miyuki Koiso, The stability and the vision number of surfaces with constant mean curvature, Bulletin of Kyoto University of Education, Ser.B, 92, 1-11, Vol.92, pp.1-11, 1998.03. |

38. | Miyuki Koiso, Applications of the maximum principle to surfaces with constant mean curvature, 京都大学数理解析研究所講究録, 995, 1-6, 1997.05. |

39. | Miyuki Koiso, A generalization of Steiner symmetrization for immersed surfaces and its applications, Manuscripta Mathematica, 87, 3, 311-325, 1995.07. |

40. | Miyuki Koiso, A uniqueness result for minimal surfaces in S^3, Advanced Studies in Pure Mathematics, 22, 117-122, 1993.09. |

41. | Miyuki Koiso, Symmetry of surfaces of constant mean curvature with symmetric boundary, Geometry and global analysis (Sendai, 1993.7), Tohoku University, Sendai, Japan., 239-248, 1993.07. |

42. | Miyuki Koiso, On the uniqueness for hypersurfaces with constant mean curvature in R^(n+1) bounded by a round (n-1)-sphere, "The Problem of Plateau" (ed. by Th.M.Rassias), World Scientific, 129-13, 1992.05. |

43. | Miyuki Koiso, Function theoretic and functional analytic methods for minimal surfaces, Reports on Global Analysis, 13, Ko 0-Ko 69, 1989.10. |

44. | Miyuki Koiso, The uniqueness for minimal surfaces in S^3, Manuscripta Mathematica, 63, 2, 193-207, 1989.02. |

45. | Miyuki Koiso, Perturbation of surfaces with constant mean curvature, 京都大学数理解析研究所講究録, 600, 135-150, 1986.10. |

46. | Miyuki Koiso, ymmetry of hypersurfaces of constant mean curvature with symmetric boundary, Mathematische Zeitschrift, 191, 4, 567-574, 1986.04. |

47. | Miyuki Koiso, On the stability of minimal surfaces in R^3, Journal of the Mathematical Society of Japan, 36, 3, 523-541, 1984.07. |

48. | Miyuki Koiso, The stability and the Gauss map of minimal surfaces in R^3, Lecture Notes in Mathematics, Springer-Verlag, 1090, 77-92, 1984.01. |

49. | Miyuki Koiso, On the space of minimal surfaces with boundaries, Osaka Journal of Mathematics, 20, 4, 911-921, 1983.12. |

50. | Miyuki Koiso, On the finite solvability of Plateau's problem for extreme curves, Osaka Journal of Mathematics, 20, 1, 177-183, 1983.02. |

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