Miyuki Koiso | Last modified date：2018.08.17 |

Professor /
Division of Fundamental mathematics /
Institute of Mathematics for Industry

**Presentations**

1. | Miyuki Koiso, Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature, International Workshop "Geometry of Submanifolds and Integrable Systems", 2018.03, [URL]. |

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4. | Miyuki Koiso, Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-shrinkers of anisotropic mean curvature flow, Workshop "Minimal Surfaces and Related Topics", 2018.01, [URL], We study a variational problem for surfaces 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 surface, which was introduced to model the surface tension of a small crystal. The minimizer of such an energy among all closed surfaces enclosing the same volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium surfaces of this energy for volume-preserving variations are generalizations of the round sphere and constant mean curvature surfaces, respectively. However, they are not smooth in general. In this talk, we show that, if the energy density function is three times continuously differentiable and convex, then any closed stable equilibrium surface is a rescaling of the Wulff shape. Moreover, we show that, there exists a non-convex energy density function such that there exist closed embedded equilibrium surfaces with genus zero which are not (any homothety of) the Wulff shape. This gives also closed embedded self-similar shrinking solutions with genus zero of the anisotropic mean curvature flow other than the Wulff shape. These concepts and results are naturally generalized to higher dimensions.. |

5. | Miyuki Koiso， Convex and non-convex equilibria for anisotropic surface energy, 2nd Workshop on Convexity in Miyazaki， October 7-9, 2017, Miyazaki University., [URL]. |

6. | Miyuki Koiso, Non-uniqueness of closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-shrinkers of anisotropic mean curvature flow, The Third Japanese-Spanish Workshop on Differential Geometry, 2017.09, [URL]. |

7. | Miyuki Koiso, Non-uniqueness of closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-shrinkers of anisotropic mean curvature flow, The Last 60 Years of Mathematical Fluid Mechanics: Longstanding Problems and New Perspectives: In Honor of Professors Robert Finn and Vsevolod Solonnikov, 2017.08, [URL], We study variational problems for surfaces in the euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density which depends on the surface normal over the considered surface. It was first introduced by Gibbs to model the equilibrium shape of a small crystal. If the energy density is constant one, the anisotropic surface energy is the usual area of the surface. The minimizer of an anisotropic surface energy among all closed surfaces enclosing the same volume is unique (up to translations) and it is called the Wulff shape. Equilibrium surfaces of a given anisotropic surface energy functional for volume-preserving variations are called surfaces with constant anisotropic mean curvature (CAMC surfaces). In general, the Wulff shape and CAMC surfaces are not smooth. If the energy density satisfies the so-called convexity condition, the Wulff shape is a smooth convex surface and closed embedded CAMC surfaces are only homotheties of the Wulff shape. In this talk, we show that if the convexity condition is not satisfied, such a uniqueness result is not always true, and also the uniqueness for self-shrinkers with genus zero for anisotropic mean curvature flow does not hold in general. These concepts and results are naturally generalized to higher dimensions.. |

8. | Miyuki Koiso, Geometry of anisotropic surface energy, The 13th annual international conference of KWMS (Korean Women in Mathematical Science), 2017.06, [URL], One of the most important subjects in geometry is variational problem. In this talk, we study variational problems for surfaces in the euclidean space with an anisotropic surface energy. An anisotropic surface energy is the integral of an energy density which depends on the surface normal over the considered surface. It was first introduced by Gibbs to model the equilibrium shape of a small crystal. If the energy density is constant one, the anisotropic surface energy is the usual area of the surface. The minimizer of an anisotropic surface energy among all closed surfaces enclosing the same volume is unique (up to translations) and it is called the Wulff shape. Equilibrium surfaces of a given anisotropic surface energy functional for volume-preserving variations are called surfaces with constant anisotropic mean curvature (CAMC surfaces). In general, the Wulff shape and CAMC surfaces are not smooth. Around each regular (smooth) point, they are graphs of solutions of a second order quasilinear elliptic partial differential equation. These concepts are naturally generalized to higher dimensions, and they have many applications inside and outside mathematics. In this talk, we give fundamental geometric and analytic properties of CAMC hypersurfaces and recent progress in the research on the uniqueness of closed CAMC hypersurfaces with and without singularities.. |

9. | Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, OIST Mini Symposium: Viscoelasticity and Dissipative Dynamics of Rods and Membranes, 2017.03, [URL]. |

10. | Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, Southeast Geometry Seminar, 2017.02, [URL]. |

11. | Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, 第10回GEOSOCKセミナー:「曲面と幾何学的変分問題」 （阪大-阪市大‐神戸大-九大合同幾何学セミナー）, 2016.12, [URL]. |

12. | Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, Workshop "Differential Geometry, Lie Theory and Low-Dimensional Topology", 2016.12, [URL]. |

13. | Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, Workshop on "Geometric Inequalities on Riemannian Manifolds", 2016.11, [URL], A surface with constant mean curvature (CMC surface) is an equilibrium surface of the area functional among surfaces which enclose the same volume and satisfy given boundary conditions. A CMC surface is said to be stable if the second variation of the area is nonnegative for all volume-preserving variations. In this talk we first give criteria for stability of CMC surfaces in R^3. We also give a sufficient condition for the existence of smooth bifurcation branches of fixed boundary CMC surfaces, and we discuss stability/instability issues for the surfaces in bifurcating branches. By applying our theory, we determine the stability/instability of some explicit examples of CMC surfaces.. |

14. | Miyuki Koiso, Anisotropic surface energy and surfaces with edges, NCTS Differential Geometry Seminar, 2016.05. |

15. | Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, Twelfth Taiwan Geometry Symposium, 2016.05. |

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17. | 小磯 深幸, Anisotropic surface energy and surfaces with edges, Workshop "Transformations and Singularities", 2016.02, [URL], 曲面の各点の向きに依存して決まる非等方的表面エネルギーは結晶の表面張力の数理モデルを与える．このようなエネルギー汎関数の停留点について，存在と一意性，エネルギーの第1及び第2変分とエネルギー極小性，曲率を始めとする幾何的性質と自然に現れる特異点について紹介した．. |

18. | 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.. |

19. | 小磯 深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in the three-dimensional Euclidean space, Workshop "Geometric Analysis in Geometry and Topology 2015", 2015.11, [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.. |

20. | 小磯 深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in the three-dimensional Euclidean space, 8th International Congress on Industrial and Applied Mathematics, 2015.08, [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.. |

21. | 小磯 深幸, Stability analysis for surfaces with constant mean curvature, Summer School on multiscale and geometric analysis, 2015.07, [URL]. |

22. | 小磯 深幸, Stability and bifurcation for surfaces with constant mean curvature, Singularities in Generic Geometry and its Applications Kobe-Kyoto 2015 (Valencia IV), 2015.06, [URL]. |

23. | 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.. |

24. | 小磯 深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in R^3, セミナー, 2015.04. |

25. | 小磯 深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in R^3, Colloquium, 2014.11. |

26. | 小磯 深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in R^3, 5th International Workshop on Differential Geometry and Analysis, 2014.06. |

27. | 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). . |

28. | 小磯 深幸, Balancing formula for immersed hypersurfaces with constant anisotropic mean curvature and its applications, 名城大学幾何学研究集会 "Progress of geometric structures on manifolds", 2014.03, We derive a ``Balancing Formula'' for immersed hypersurfaces with constant anisotropic mean curvature (CAMC) in the euclidean space. By using this formula, we prove that, for an axially symmetric anisotropic surface energy, any embedded CAMC hypersurface spanned by a sphere is part of a homothety of the Wulff shape under a certain natural assumption for the hypersurface. . |

29. | 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).. |

30. | 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.. |

31. | 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. . |

32. | 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.. |

33. | 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). . |

34. | 小磯 深幸, Stability of hypersurfaces with constant mean curvature and applications to isoperimetric problems, 偏微分方程式の解の幾何(RIMS 研究集会), 2012.11, We give some criteria for the stability for surfaces with constant mean curvature. They are given by the properties of the eigenvalues and the eigenfunctions of the eigenvalue problems associated with the second variation of the area. Especially, a new criterion for the stability is given by using bifurcations. We apply our methods to the isoperimetric problem in S^1 times R^n. . |

35. | 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.. |

36. | 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]. |

37. | 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]. |

38. | 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]. |

39. | Miyuki Koiso, Geometric variational problems and bifurcation theory, Colloquium (Department of Mathematical Sciences, Tsinghua University, P. R. China), September 22, 2011. . |

40. | Miyuki Koiso, Existence and uniqueness for compact stable surfaces of constant anisotropic mean curvature with prescribed boundary condition, Geometry Seminar, (Department of Mathematical Sciences, Tsinghua University, P. R. China), September 17, 2011.. |

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42. | 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]. |

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

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

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

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

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

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53. | Stability and uniqueness for surfaces with constant anisotropic mean curvature, [URL]. |

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