Kyushu University Academic Staff Educational and Research Activities Database
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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
1. Miyuki Koiso, Bennett Palmer, Higher order variations of constant mean curvature surfaces, Calculus of Variations and PDE's, 10.1007/s00526-017-1246-1, 2018.12, 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..
2. 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..
3. 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 $\partial\Sigma$..
4. 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..
5. 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。.
6. 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.
7. 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.
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