九州大学 研究者情報
発表一覧
小磯 深幸(こいそ みゆき) データ更新日:2019.10.28

教授 /  マス・フォア・インダストリ研究所 基礎理論研究部門


学会発表等
1. Miyuki Koiso, Geometry of anisotropic surface energy, KU-NTNU Joint Forum, 2019.05, [URL].
2. Miyuki Koiso, Variational problems of anisotropic surface energy for hypersurfaces with singular points, AMS Spring Central and Western Joint Sectional Meeting, 2019.03, [URL].
3. 小磯深幸, 特異点を持つ非等方的平均曲率一定閉超曲面の一意性について, 淡路島幾何学研究集会2019, 2019.01.
4. 小磯 深幸, 特異点を許容する超曲面に対する非等方的エネルギーの変分問題, 第20回特異点研究会「特異点と時空、および関連する物理」, 2019.01, [URL].
5. Miyuki Koiso, Towards crystalline variational problems from elliptic variational problems, Introductory workshop on discrete differential geometry, 2019.01, [URL].
6. Miyuki Koiso, Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature and applications to anisotropic mean curvature flow, Conference "Analysis and Geometry in Minimal Surface Theory", 2018.12, [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 give a suitable formulation of piecewise-smooth hypersurfaces and discuss geometry of equilibrium hypersurfaces. Especially, we give recent results on the uniqueness for closed equilibria and their applications to anisotropic mean curvature flow..
7. Miyuki Koiso, Uniqueness problems for closed non-smooth hypersurfaces with constant anisotropic mean curvature and applications to anisotropic mean curvature flow, Department of Mathematics Colloquium (NTNU & NCTS Differential geometry Seminar), 2018.12, [URL].
8. 小磯 深幸, Geometry of anisotropic surface energy, 第2回 岡潔女性数学者セミナー, 2018.11, [URL].
9. Miyuki Koiso, Anisotropic surface energy and crystalline variational problems, colloquium,TUWien, 2018.10.
10. Miyuki Koiso, Crystalline variational problem and applications to capillary problems, 7th International Conference on Mathematical Modeling in Physical Sciences, 2018.08, 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 not smooth in general. In this paper, we give a formulation of piecewise-smooth hypersurfaces and discuss geometry of equilibrium hypersurfaces in the Euclidean space of general dimension. Especially, we give uniqueness and non-uniqueness results for closed equilibria. We also mention applications to anisotropic mean curvature flow and to capillary problems..
11. Miyuki Koiso, Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature, The 11th Mathematical Society of Japan Seasonal Institute (MSJ-SI): The Role of Metrics in the Theory of Partial Differential Equations, 2018.07, [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 talk 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..
12. Miyuki Koiso, Geometry of anisotropic surface energy and crystalline variational problem, Mini-Workshop on Geometry and Mathematical Science, 2018.07, [URL].
13. 小磯 深幸, Uniqueness for closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature, 京都大学数理解析研究所(RIMS)共同研究・公開型「偏微分方程式の解の形状解析」, 2018.06, [URL].
14. 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].
15. 小磯深幸, 特異点を持つ非等方的平均曲率一定閉超曲面の非一意性と非等方的平均曲率流方程式への応用, 日本数学会2018年度年会, 2018.03, [URL].
16. 小磯深幸, 特異点を持つ安定な非等方的平均曲率一定閉超曲面の一意性, 日本数学会2018年度年会, 2018.03, [URL].
17. 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..
18. 小磯深幸, 滑らかな幾何学と離散幾何学をつなぐ幾何概念創造への挑戦, 淡路島幾何学研究集会2018, 2018.01.
19. 小磯深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in R^3, 研究集会「離散幾何解析とその周辺」, 2017.12, [URL].
20. 小磯深幸 , Non-uniqueness of closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-shrinkers of anisotropic mean curvature flow, 福岡大学微分幾何研究会, 2017.11, [URL].
21. 小磯深幸, Convex and non-convex equilibria for anisotropic surface energy, 2nd Workshop on Convexity in Miyazaki, 2017.10, [URL].
22. 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].
23. 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..
24. 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..
25. 小磯深幸, 非等方的表面エネルギーの幾何, 京都大学数理解析研究所(RIMS)共同研究「部分多様体論の潮流」, 2017.06, [URL].
26. 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].
27. 小磯深幸, 平均曲率一定曲面の弱安定性と高次の変分, 小磯憲史先生退職記念研究集会, 2017.03, [URL].
28. 小磯深幸, 赤嶺新太郎, 特異点を持つ曲面の幾何学及び離散幾何学とそれらのキンク変形研究への応用可能性, 日本金属学会 キンク研究会 平成28年度研究会, 2017.03, [URL], 金属の折れ曲がりの微細構造は,幾何学における滑らかな曲面の平行曲面族とそれにより生じる特異点を想起させる.本講演では,平行曲面族及び曲面の特異点の分類についての数学理論を解説し,キンク変形研究への応用可能性を探る.さらに,曲線論・曲面論を物理現象解明に応用する際にしばしば有用である離散幾何学の考え方について解説する.
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29. Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, Southeast Geometry Seminar, 2017.02, [URL].
30. 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].
31. Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, Workshop "Differential Geometry, Lie Theory and Low-Dimensional Topology", 2016.12, [URL].
32. 小磯深幸, Stability of constant mean curvature surfaces and higher order variations, 福岡大学微分幾何研究会, 2016.11, [URL].
33. 小磯深幸, 平均曲率一定曲面に対する分岐理論とDelaunay曲面の安定性判定への応用, 研究集会「多様体上の微分方程式」金沢シリーズ第15回, 2016.11, [URL].
34. 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..
35. 小磯深幸, 曲面に対する非等方的エネルギーの幾何 --- 結晶の数理モデルの幾何学 ---, 日本機械学会2016年度年次大会, 2016.09.
36. 小磯深幸, 曲面の変分問題の解の安定性解析とコンピュータによる曲面表現, 九大IMI 2016年度 短期共同利用「三次元幾何モデリング評価手法の提案とソフトウェア開発」, 2016.08.
37. Miyuki Koiso, Anisotropic surface energy and surfaces with edges, NCTS Differential Geometry Seminar, 2016.05.
38. Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature, Twelfth Taiwan Geometry Symposium, 2016.05.
39. 小磯深幸, 3次元ユークリッド空間内の三重周期極小曲面(TPMS)の剛性・分岐・共連続構造への応用, CAMM(コンピュータによる材料開発・物質設計を考える会)フォーラム, 2016.04.
40. 小磯 深幸, 曲面に対する非等方的エネルギーの幾何, 日本数学会2016年度年会 応用数学分科会, 2016.03, [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.
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41. 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..
42. 小磯 深幸, 3次元ローレンツ多様体内の有界な平均曲率を持つ混合型曲面(平均曲率方程式が楕円型,双曲型,放物型のすべての部分を持つ曲面)について, 九州関数方程式セミナー, 2016.01, [URL], We show 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 show the existence of such surfaces with non-vanishing mean curvature and investigate their properties..
43. 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..
44. 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..
45. 小磯 深幸, Local structure of the space of all triply periodic minimal surfaces in R^3, 2015年度福岡大学微分幾何研究会(Geometry and Analysis), 2015.10, [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..
46. 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..
47. 小磯 深幸, Stability analysis for surfaces with constant mean curvature, Summer School on multiscale and geometric analysis, 2015.07, [URL].
48. 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].
49. 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..
50. 小磯 深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in R^3, セミナー, 2015.04.
51. 小磯 深幸, 3次元ユークリッド空間内の三重周期極小曲面の剛性と分岐, 金沢大学理学談話会, 2015.02.
52. 小磯 深幸, 効率の良い形は美しいか? --- 曲面の変分問題と応用 ---, 九州大学テクノロジーフォーラム2014, 2014.12.
53. 小磯 深幸, On bifurcation and local rigidity of triply periodic minimal surfaces in R^3, Colloquium, 2014.11.
54. 小磯 深幸, 3次元ユークリッド空間内の三重周期極小曲面の剛性・分岐・共
連続構造への応用, 日本応用数理学会2014年度年会, 2014.09.
55. 小磯 深幸, 3次元ユークリッド空間内の三重周期極小曲面の局所剛性と分岐について, 近畿大学理工学部理学科物理コース ソフトマター研究室セミナー, 2014.09.
56. 小磯 深幸, 3次元ユークリッド空間内の三重周期極小曲面の剛性と分岐について, 第61回幾何学シンポジウム, 2014.08.
57. 小磯 深幸, 3次元ユークリッド空間内の三重周期極小極小曲面の剛性と分岐について, 九州大学数理学研究院談話会, 2014.06.
58. 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.
59. 小磯 深幸, 3次元ユークリッド空間内の三重周期極小極小曲面の剛性と分岐について, 第7回OCU48セミナー, 2014.04.
60. 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). .
61. 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, [URL], 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.
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62. 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)..
63. 小磯 深幸, 曲面の変分問題の解の安定性 --- 判定法と応用.極小曲面と平均曲率一定曲面を中心に ---, 山口幾何学研究集会2013 --- 進化する曲面論 ---, 2013.12, 曲面の変分問題の解の安定性についての基礎概念から最新の結果までを,極小曲面と平均曲率一定曲面に関するものを中心に解説した..
64. 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..
65. 小磯 深幸, 非等方的平均曲率一定曲面に対するバランス公式とその応用, 福岡大学幾何学研究会, 2013.11, 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の臨界点である非等方的平均曲率一定超曲面(CAMC超曲面)に対するバランス公式を紹介する.さらに,その応用として,軸対称なエネルギー汎関数に対し,与えられた境界を張るCAMC超曲面に対する一意性定理を得る..
66. 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. .
67. 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..
68. 小磯 深幸, 平均曲率一定曲面に対する自由境界問題と解の安定性, 東京工業大学 数理解析セミナー, 2013.07, [URL], 3次元ユークリッド空間内与えられた二平面(以下では支持曲面と呼ぶ)上に自由境界をもち, これらの平面で囲まれる領域に埋め込まれたコンパクト曲面全体を考える. 「囲む体積」を保つ変分に対する「総エネルギー=面積+自由境界での濡れエネルギー」の臨界点は, 支持平面と成す角度が一定であるような平均曲率一定曲面となる. 臨界点は,「囲む体積」を保つ変分に対する総エネルギーの第二変分が非負の時に安定であると呼ばれる. 本講演では,安定解の決定のための方法と結果,及び, 支持曲面の変動に対する解の不連続性について最近得られた結果をご報告する. なお,本講演は,Bennett Palmer氏(米国・アイダホ州立大学), Jaigyoung Choe氏(韓国・KIAS)との共同研究に基づく..
69. 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). .
70. 小磯 深幸, 非等方的平均曲率一定曲面の幾何, 熊本大学大学院自然科学研究科数学専攻プロジェクトゼミナール, 2013.01, [URL], ユークリッド空間内の(超)曲面に対し,その各点の向きに依存するエネルギー密度の曲面上での積分を,この曲面の非等方的エネルギーという.非等方的エネルギーは面積の一般化であり,たとえば結晶の表面張力の数理モデルを与える.曲面が囲む体積を変えない変分に対する非等方的エネルギーの臨界点は,非等方的平均曲率が至る所一定の曲面(以下ではCAMC曲面と呼ぶ)となる.非等方的平均曲率は,曲面の各点での向きに依存する重みの付いた曲率である.CAMC曲面は,その特別な場合として,ユークリッド空間及びローレンツ空間内の極小(大)曲面や平均曲率一定曲面,調和関数のグラフ等を含む.本講演では,CAMC曲面についての基本概念及び講演者による最近の研究について解説する.第1部では,非等方的平均曲率を導出し,非等方的エネルギー及びCAMC曲面のさまざまな例を紹介する.第2部では,エネルギー汎関数の第2変分を導出し,それが常に非負という意味でCAMC曲面の安定性を定義する.特に,CAMC曲面は,囲む体積を変えない変分に対する非等方的エネルギーの極小値を与えるならば,安定である.さらに,安定なCAMC曲面の非等方的Gauss写像(単位法ベクトル場の一般化)の一般化ディリクレエネルギー最小性,円を境界に持つ安定で種数0のCAMC曲面に対する一意性定理を紹介する..
71. 小磯 深幸, 非等方的平均曲率一定曲面の幾何, 研究集会「界面の数理と幾何解析」, 2012.11, [URL], 曲面の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)を非等方的エネルギーという.これは結晶やある種の液晶のエネルギーの数理モデルを与える.非等方的エネルギーの臨界点は,極小曲面や平均曲率一定曲面の一般化を与える.これらに関する研究について紹介した..
72. 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, [URL], 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. .
73. 小磯 深幸, S^1 × R^n上の等周問題 --- 分岐理論の応用 ---, 福岡大学幾何学研究会, 2012.11, [URL], N=S^1 × R^n において,同じ(n+1)次元体積を囲むコンパクトな閉超曲面の中でのn次元体積の臨界点の安定性を決定するという問題を考える.用いる方法は,さまざまな対称化法,常微分方程式の解の解析,平均曲率一定(CMC)超曲面の安定性の判定法(第2変分に付随する固有値問題),CMC超曲面に対する境界値問題の解の分岐の存在条件と分岐前後の解の安定性の判定,並びに,数値計算の援用である..
74. 小磯 深幸, Stability of hypersurfaces with constant mean curvature and applications to isoperimetric problems, 大阪市立大学数学研究所 談話会 特別企画 --- A Big Wave Special Colloquium ---, 2012.09, [URL], 平均曲率一定超曲面(CMC超曲面)は,曲面が「囲む体積」を保つ変分に対する面積の臨界点であり,このような変分に対する面積の第二変分が非負の時に「安定である」と言われる.本講演では,自由境界あるいは固定境界を持つCMC超曲面の安定性を判定するいくつかの方法について延べ,それらの応用として,直積多様体(S^1)×(R^n)における等周問題の解を決定する.安定性の判定に際しては,ヒルベルト空間あるいはバナッハ空間上の汎関数の臨界点についての一般論を応用するが,「直接適用する」ことができない理由についても説明したい.さらに,理論の一般化や応用例についても述べたい..
75. 小磯 深幸, 曲線と曲面の微分幾何学の基礎 --- 諸分野への応用を念頭に ---, CREST Workshop 生体形状モデリングと幾何, 2012.07, [URL], 曲線と曲面の曲率について説明し、自然現象解明への応用例と今後の展望について述べた。.
76. 小磯 深幸, Geometry of isoperimetric-type problems modeled on interfaces on micrometre scale, 立教大学・理論物理学研究室・宇宙コロキウム, 2012.07, [URL], 曲面に対する境界条件が与えられた時,曲面が囲む体積を保つ変分に対する面積 の臨界点は平均曲率一定曲面となる.臨界点が面積の極小値を与える時,安定で あるという.ナノあるいはミクロスケールの液体は,安定な平均曲率一定曲面と みなすことによりその形状の説明がつくことがある.本講演では,平均曲率一定 曲面の安定性の判定条件,解の分岐が起こるための十分条件,分岐前後の解の安 定性を判定する方法を与え,物理現象に動機を得た等周問題への応用や,より一 般のエネルギー汎関数の臨界点に対する一般化について述べる..
77. 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..
78. 小磯 深幸, 平均曲率一定曲面の安定性と等周問題への応用, 金沢大学理学部数理学談話会, 2012.06, [URL], 平均曲率一定曲面の安定性(対応する汎関数の極小値を与えるか否か)の判定条件,解の分岐が起こるための十分条件,分岐前後の解の安定性を判定する方法を与え,等周問題への応用について述べた..
79. 小磯 深幸, 平均曲率一定曲面の分岐と安定性及びその一般化, CRESTセミナー(JST/CREST 小谷チーム「離散幾何学から提案する新物質創成・物性発現の解明」), 2012.04, [URL], 囲む体積を保つ変分に対する面積の臨界点である平均曲率一定曲面や, その一般化について,安定性(対応する汎関数の極小値を与えるか否か)の判定条件,解の分岐が起こるための十分条件,分岐前後の解の安定性を判定する方法, pitchfork分岐や対称性の崩壊現象が生じるための条件を与える. さらに,それらをRiemann多様体上の等周問題や,物理現象と関連の深いいくつかの(自由あるいは固定)境界値問題に応用する. .
80. 小磯深幸, 等周問題型変分問題の幾何解析, 日本数学会2012年度年会, 2012.03, [URL], 囲む体積を保つ変分に対する面積汎関数の臨界点である平均曲率一定曲面や,その一般化について,安定性(対応する汎関数の極小値を与えるか否か)の判定条件,解の分岐が起こるための十分条件,分岐前後の解の安定性を判定する方法を与え,pitchfork分岐や対称性の崩壊現象が生じることを見る..
81. Miyuki Koiso, Bifurcation and stability for solutions of isoperimetric problems, Workshop "Isoperimetric problems, space-filling, and soap bubble geometry", 2012.03, [URL], A surface with constant mean curvature (CMC surface) is an equilibrium surface of the area functional among surfaces which enclose the same volume with given boundary condition. A CMC surface is said to be stable if the second variation of the area is nonnegative for all volume-preserving variations which satisfy the boundary condition. Choosing the mean curvature H or the volume V enclosed by the surface as parameter, we construct conditions under which a pitchfork bifurcation occurs. We apply our results to several interesting isoperimetric problems with free or fixed boundary conditions in the euclidean space. We also generalize the results to more general variational problems with constraint in higher dimensional spaces..
82. 小磯深幸, 幾何学的変分問題の大域解析とナノ-ミクロ物質の形態解明への応用, 越境する数学 ~さきがけ第二期生 研究成果報告会~ , 2011.12, [URL].
83. Miyuki Koiso, Pitchfork bifurcation for hypersurfaces with constant mean curvature, The 10th Pacific Rim Geometry Conference 2011 Osaka-Fukuoka, 2011.12, [URL], We consider hypersurfaces with constant mean curvature with given boundary conditions. Choosing the mean curvature $H$ or the volume $V$ enclosed by the hypersurface as parameter, we construct conditions under which a pitchfork bifurcation occurs. We apply our results to isoperimetric problems in the Riemannian products of $S^1$ and simply connected space forms introduced by Pedrosa and Ritor\'e (1999)..
84. 小磯深幸, 非等方的平均曲率一定曲面の基礎理論, 広島幾何学研究集会 2011, 2011.10, ユークリッド空間内の非等方的平均曲率一定超曲面についての基礎概念,及び,非等方的平均曲率一定超閉曲面に対する一意性定理と未解決問題について述べた..
85. Miyuki Koiso, Geometric variational problems and bifurcation theory, Colloquium, 2011.09, In the study of geometric variational problems, it is natural to ask whether each critical point is stable (that is, the considered critical point attains a local minimum of the energy) or not. Also it is important to determine the geometric properties of solutions and to study the structure of the set of solutions. In this talk, as one of the steps to investigate these problems, we discuss stability, existence of bifurcation, and symmetry-breaking for solutions of variational problems for hypersurfaces with constraint. Although our method is sufficiently general to apply various variational problems, we mainly concentrate on hypersurfaces with constant anisotropic mean curvature in the euclidean space, which are characterized as critical points of anisotropic surface energy with volume constraint. Useful criteria for the stability and existence of bifurcation are given by the properties of eigenvalues and eigenfunctions of the eigenvalue problem associated with the second variation of the energy. We will give general methods and their applications to several concrete examples which may be interesting from both mathematical and physical point of view. .
86. Miyuki Koiso, Existence and uniqueness for compact stable surfaces of constant anisotropic mean curvature with prescribed boundary condition, Geometry Seminar, 2011.09, If we are given a variational problem for surfaces with boundary and the variational problem and the boundary of a critical surface have the same symmetry, must the critical surface also have the same symmetry? Alias-Lopez-Palmer (1999) proved that, in the case where the functional is the area, any stable constant mean curvature immersion in the three-dimensional euclidean space of a topological disc, which is bounded by a round circle, is necessarily rotationally symmetric and is hence a spherical cap or a flat disc. In this talk, we give an extension of this result to the case of surfaces with constant anisotropic mean curvature. Moreover, we discuss a similar subject for the case where there are more than one boundary components to get stable solutions with less symmetry. .
87. 小磯深幸, 非等方的平均曲率一定曲面の安定性解析と物理現象への応用, 研究集会「部分多様体幾何とリー群作用2011」 , 2011.09, [URL], A surface with constant anisotropic mean curvature (CAMC surface) is a critical point of an anisotropic surface energy with volume constraint. A CAMC surface is said to be stable if the second variation of the anisotropic surface energy is nonnegative for all "volume"-preserving variations satisfying given boundary conditions. Useful criteria for stability and existence of bifurcation are given by the eigenvalue problem associated with the second variation of the energy. We will give general methods and applications to several concrete examples which may be interesting from both mathematical and physical points of view..
88. Miyuki Koiso, Stability of surfaces with constant anisotropic mean curvature and applications to physical phenomena, III Encontro Paulista de Geometria, 2011.08, [URL], A surface with constant anisotropic mean curvature (CAMC surface) is a critical point of an anisotropic surface energy with volume constraint. CAMC surfaces are a generalization of minimal surfaces and surfaces with constant mean curvature (CMC surfaces). They are not only important in itself but also useful in studying, for example, free boundary problems for CMC and minimal surfaces. A CAMC surface is said to be stable if the second variation of the anisotropic surface energy is nonnegative for all "volume"-preserving variations satisfying given boundary conditions. Useful criteria for stability and existence of bifurcation are given by the eigenvalue problem associated with the second variation of the energy. We will give general methods and applications to several concrete examples which may be interesting from both mathematical and physical points of view..
89. 小磯深幸, 非等方的平均曲率一定曲面の幾何, 日本数学会 2011 年度年会, 2011.03.
90. Miyuki Koiso, Stability of hypersurfaces with constant anisotropic mean curvature and
its applications, Spanish-Japanese Workshop on Differential Geometry, 2011.02.
91. Miyuki Koiso, Morse index, stability, and bifurcation theory for variational problems
for hypersurfaces with constraint, The Third International Workshop on Differential Geometry, 2011.01.
92. 小磯深幸, 非等方的平均曲率一定超曲面の幾何, 大岡山談話会, 2010.12, [URL].
93. Miyuki Koiso, Geometric variational problems and bifurcation theory, 2010 Global KMS International Conference, 2010.10, [URL].
94. Miyuki Koiso, Geometry of hypersurfaces with constant anisotropic mean curvature, Workshop on Hypersurfaces Geometry and Integrable Systems, 2010.08, [URL].
95. Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature and their generalizations, 16th School of Differential Geometry, 2010.07, [URL].
96. Miyuki Koiso, Stability and bifurcation for surfaces with constant mean curvature and their generalizations, Oberwolfach workshop ``Progress in Surface Theory'', 2010.05, [URL].
97. 小磯深幸, 等周問題型変分問題の解のエネルギー極小性と分岐について, 九州関数方程式セミナー, 2010.04.
98. 小磯深幸, 非等方的平均曲率一定曲面に対する自由境界問題, 九州大学幾何学セミナー, 2010.04.
99. Miyuki Koiso, Stability and bifurcation for solutions of isoperimetric type problems, The 2nd TIMS-OCAMI Joint International Workshop on Differential Geometry and Geometric Analysis, 2010.03, [URL].
100. 小磯深幸, 曲面の変分問題の解の大域解析とその応用, 大阪市立大学数学教室談話会, 2010.02.
101. Miyuki Koiso, Geometric variational problems and bifurcation theory,, The First CREST-SBM International Conference "Random Media", 2010.01, [URL].
102. Miyuki Koiso and Bennett Palmer, An anisotropic version of Hopf’s Theorem, 東京幾何セミナー, 2009.12, [URL].
103. 小磯深幸, Variational Problems for Anisotropic Surface Energies, 第34回偏微分方程式論札幌シンポジウム, 2009.08, [URL].
104. Miyuki Koiso, Uniqueness for "small" CMC (constant mean curvature) surfaces with boundary, 大阪市立大学微分幾何学セミナー, 2009.07.
105. Miyuki Koiso, Variational problems for anisotropic surface energies, PDE実解析研究会, 2009.06.
106. Miyuki Koiso, Stability of hypersurfaces with constant mean curvature, Differentail Geometry with Mira, 2009.04.
107. Miyuki Koiso, Stability of hypersurfaces with constant mean curvature, International Conference "Global Analysis and Differential Geometry", 2009.03, [URL].
108. Miyuki Koiso, A free boundary problem for surfaces with constant anisotropic mean curvature and related topics, Geometry Seminar, 2009.03.
109. Miyuki Koiso, Geometry and stability of rotationally symmetric hypersurfaces with constant mean curvature, Integrable systems, Geometry and Visualization 2008, 2008.12.
110. Miyuki Koiso, Anisotropic Gauss map of surfaces, One Day workshop on Differential Geometry, 2008.11, [URL].
111. 小磯深幸, Anisotropic surface energy and anisotropic Gauss map of surfaces, 東北大学幾何セミナー, 2008.10, [URL].
112. Miyuki Koiso, Gauss map and anisotropic Gauss map of surfaces with constant anisotropic mean curvature, Colloquium, Tsinghua University, 2008.09.
113. Miyuki Koiso, Gauss map and anisotropic Gauss map of surfaces with constant anisotropic mean curvature, Colloquium, Tsinghua University, 2008.09.
114. Miyuki Koiso, A free boundary problem for surfaces with constant anisotropic mean curvature, CONFERENCE Differential Geometry, 2008.06, [URL].
115. 小磯深幸, 非等方的平均曲率一定曲面の安定性と一意性について, 東京大学・数理科学研究科・応用解析セミナー, 2008.05, [URL].
116. 小磯深幸, 非等方的平均曲率一定曲面に対する自由境界問題, 筑波大学微分幾何学火曜セミナー, 2008.02, [URL].
117. 小磯深幸, 非等方的平均曲率一定曲面に対する自由境界問題, 研究集会「リーマン幾何と幾何解析」, 2008.02, [URL].
118. Miyuki Koiso, A free boundary problem for surfaces with constant anisotropic mean curvature, The 3rd Geometry Conference for Friendship of Japan and China , 2008.01, [URL].

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