Differential geometry of submanifolds and its related topics : proceedings of the International Workshop in honor of S. Maeda's 60th birthday : Saga University, Saga, Japan, 4-6 August 2012

This Volume Is A Compilation Of Papers Presented At The Conference On Differential Geometry, In Particular, Minimal Surfaces, Real Hypersurfaces Of A Non-flat Complex Space Form, Submanifolds Of Symmetric Spaces And Curve Theory. It Also Contains New Results Or Brief Surveys In These Areas. This Volume Provides Fundamental Knowledge To Readers (such As Differential Geometers) Who Are Interested In The Theory Of Real Hypersurfaces In A Non-flat Complex Space Form. Homogeneous Submanifolds And Homogeneous Curves In Space Forms / S. Maeda -- Injectivity Property Of Regular Curves And A Sphere Theorem / O. Kobayashi -- A Family Of Complete Minimal Surfaces Of Finite Total Curvature With Two Ends / S. Fujimori And T. Shoda -- Minimal Surfaces In The Anti-de Sitter Spacetime / T. Ichiyama And S. Udagawa -- Extrinsic Circular Trajectories On Geodesic Spheres In A Complex Projective Space / T. Adachi -- Geometry Of Certain Lagrangian Submanifolds In Hermitian Symmetric Spaces / Y. Ohnita -- Some Real Hypersurfaces Of Complex Projective Space / T. Hamada -- Contact Metric Hypersurfaces In Complex Space Forms / J.t. Cho And J. Inoguchi -- Non-homogeneous [eta]-einstein Real Hypersurfaces In A 2-dimensional Nonflat Complex Space Form / K. Okumura -- Sectional Curvatures Of Ruled Real Hypersurfaces In A Nonflat Complex Space Form / H. Tanabe And S. Maeda --^ Totally Geodesic Kähler Immersions Into A Complex Space Form, And A Non-existence Theorem For Hessian Metrics Of Positive Constant Hessian Sectional Curvature / T. Noda And N. Boumuki -- Archimedean Theorems And W-curves / D.-s. Kim And Y.h. Kim -- On The Construction Of Cohomogeneity One Special Lagrangian Submanifolds In The Cotangent Bundle Of The Sphere / K. Hashimoto -- Self-shrinkers Of The Mean Curvature Flow / Q.-m. Cheng And Y. Peng -- Spectrum Of Poly-laplacian And Fractional Laplacian / L. Zeng -- Flat Centroaffine Surfaces With Non-semisimple Tchebychev Operator / A. Fujioka -- The Total Absolute Curvature Of Open Curves In E[superscript N] / K. Enomoto And J. Itoh -- Antipodal Sets Of Compact Symmetric Spaces And The Intersection Of Totally Geodesic Submanifolds / M.s. Tanaka -- A Note On Symmetric Triad And Hermann Action / O. Ikawa -- Some Topics Of Homogeneous Submanifolds In Complex Hyperbolic Spaces / T. Hashinaga, A. Kubo And H. Tamaru --^ Austere Hypersurfaces In 5-sphere And Real Hypersurfaces In Complex Projective Plane / J.t. Cho And M. Kimura -- On The Minimality Of Normal Bundles In The Tangent Bundles Over The Complex Space Forms / T. Kajigaya -- Over-determined Systems On Surfaces / N. Ando. Editors, Sadahiro Maeda, Yoshihiro Ohnita, Qing-ming Cheng. The Workshop On Differential Geometry Of Submanifolds And Its Related Topics Was Held At Saga University, Saga City, Japan, During The Period Of 4-6 August, 2012 In Honor Of Professor Sadahiro Maeda (saga University) For His 60th Birthday, Which Was Supported By The Warm Hospitality Of Professor Q.m. Cheng (fukuoka University). Includes Bibliographical References And Index. CONTENTS Foreword Preface Organizing and Scientific Advisory Committees Group Photo Presentations Homogeneous submanifolds and homogeneous curves in space forms S. Maeda 1. Homogeneous real hypersurfaces in a nonflat complex space form 2. Characterizations of isoparametric hypersurfaces in asphere 3. Homogeneous curves in a nonflat complex space form 4. Parallel immersions of real space forms into real spaceforms References Injectivity property of regular curves and a sphere theorem O. Kobayashi 1. Introduction 2. Proof of theorem 3. Concluding remarks References A family of complete minimal surfaces of finite total curvature with two ends S. Fujimori and T. Shoda 1. Introduction Acknowledgments 2. Construction 3. Further progress 4. Remaining problems 4.1. The case odd and greater than 1 4.2. Existence of non-orientable minimal surfaces References Minimal surfaces in the anti-de Sitter spacetime T. Ichiyama and S. Udagawa 1. Introduction 2. anti-de Sitter spacetime 3. Oriented spacelike surfaces 4. A special solution of sinh-Gordon equation 5. Extended framing and spectral curves 6. Twisted loop group and twisted loop algebra 7. An example of oriented spacelike minimal surface in H31 8. Problems References Extrinsic circular trajectories on geodesic spheres in acomplex projective space T. Adachi 1. Introduction 2. Sasakian magnetic fields on geodesic spheres 3. Extrinsic circular trajectories on geodesic spheres 4. Moduli space of extrinsic circular trajectories 5. Other trajectories on geodesic spheres References Geometry of certain Lagrangian submanifoldsin Hermitian symmetric spaces Y. Ohnita Introduction 1. Lagrangian submanifolds in Kahler manifolds 2. Homogeneous Lagrangian submanifolds in complex projective spaces 3. Homogeneous Lagrangian submanifolds in complex hyperquadrics and hypersurface geometry in spheres 4. Construction of Lagrangian submanifolds in CPn+1 froma Lagrangian submanifold in Qn(C) 5. Construction of Lagrangian submanifolds in Qn+1(C)from a Lagrangian submanifold in Qn(C) 6. Construction of Lagrangian submanifolds in CPn+1:From a Lagrangian submanifold in CPn References Some real hypersurfaces of complex projective space T. Hamada 1. Introduction 2. Preliminaries 3. Proof of the theorem References Contact metric hypersurfaces in complex space forms J. T. Cho and J. Inoguchi Introduction 1. Contact metric manifolds 2. Real hypersurfaces 3. Real hypersurfaces in complex space forms References Non-homogeneous -Einstein real hypersurfaces in a 2-dimensional nonflat complex space form K. Okumura 1. Introduction 2. Fundamental materials, Hopf hypersurfaces and homogeneous real hyperusurfaces in Mn(c) 3. Ruled real hypersurfaces in Mn(c) 4. Conditions on the shape operator 5. Conditions on the Ricci tensor and the *-Ricci tensor 6. Main theorem References Sectional curvatures of ruled real hypersurfaces in a nonflat complex space form H. Tanabe and S. Maeda 1. Introduction 2. Ruled real hypersurfaces in Mn(c) 3. Main result References Totally geodesic Kahler immersions into a complex space form, and a non-existence theorem for Hessian metrics of positive constant Hessian sectional curvature T. Noda and N. Boumuki 1. Introduction and the main result 2. Totally geodesic Kahler immersions into a complex space form 3. Proof of Theorem 1.1 3.1. Preliminary to the proof of Theorem 1.1 3.2. Proof of Theorem 1.1 4. Remarks Appendix: A topic related to our paper References Archimedean theorems and W-curves D.-S. Kim and Y. H. Kim 1. Introduction 2. Chord property and W-curves 3. Parabolas 4. Surface property, spheres and paraboloids References On the construction of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle of the sphere K. Hashimoto 1. Introduction 2. g = 3 2.1. (G,K) = (SU(3), SO(3)) 2.2. (G,K) = (SU(3) × SU(3), SU(3)) 2.3. (G,K) = (SU(6), Sp(3)) 3. g = 4 3.1. (G,K) = (SO(m + 2), SO(2) × SO(m)) 3.2. (G,K) = (SU(m + 2), S(U(2) × U(m))) 3.3. (G,K) = (Sp(m+ 2), Sp(2) × Sp(m)) 3.4. (G,K) = (SO(5) × SO(5), SO(5)) 3.5. (G,K) = (SO(10),U(5)) Acknowledgments References Self-shrinkers of the mean curvature flow Q.-M. Cheng and Y. Peng 1. Introduction 1.1. The mean curvature flow 1.2. Characterizations of self-shrinkers 1.3. Examples of self-shrinkers 2. Complete self-shrinkers in Rn+1 2.1. Compact self-shrinkers in Rn+1 2.2. Self-shrinkers with polynomial volume growth 2.3. Without condition of polynomial volume growth 3. Complete self-shrinkers in Rn+p References Spectrum of poly-Laplacian and fractional Laplacian L. Zeng 1. Eigenvalue Problems 2. Some Technical Lemmas 3. Symmetric Rearrangements and Related Properties 4. Lower Bounds of Eigenvalues of Poly-Laplacian 5. Lower Bounds of Eigenvalues of Fractional Laplacian 6. Improvement of the Second Term 7. Lower Bounds with a Correction Term References Flat centroaffine surfaces with non-semisimple Tchebychev operator A. Fujioka 1. Introduction 2. The Gauss equations for centroaffine surfaces 3. Some invariants for centroaffine surfaces 4. Non-semisimple flat centroaffine surfaces Acknowledgements References The total absolute curvature of open curves in EN K. Enomoto and J. Itoh 1. Introduction 2. Preliminaries 3. Piecewise linear curves with two edges 4. Piecewise linear curves with n edges 5. Smooth curves References Antipodal sets of compact symmetric spaces and the intersection of totally geodesic submanifolds M. S. Tanaka 1. Introduction 2. Chen-Nagano theory 2.1. Polars and meridians 2.2. Antipodal sets and 2-numbers 3. Real forms in Hermitian symmetric spaces of compact type 3.1. Definitions and examples 3.2. Symmetric R-spaces and real forms 4. The intersection of real forms in a Hermitian symmetric space of compact type References A note on symmetric triad and Hermann action O. Ikawa 1. Introduction 2. The geometry of symmetric triad 3. Hermann actions References Some topics of homogeneous submanifolds in complex hyperbolic spaces T. Hashinaga, A. Kubo and H. Tamaru 1. Introduction 2. Preliminaries 2.1. The Cartan decomposition 2.2. The root space decomposition 2.3. The solvable model 3. Geometry of Lie hypersurfaces 3.1. Preliminaries on Lie hypersurfaces 3.2. Extrinsic geometry of Lie hypersurfaces 3.3. Curvatures of Lie hypersurfaces 3.4. Ricci soliton Lie hypersurfaces 4. Weakly reflective submanifolds 5. Cohomogeneity two actions on the complex hyperbolic plane References Austere hypersurfaces in 5-sphere and real hypersurfaces incomplex projective plane J. T. Cho and M. Kimura 1. Introduction 2. Austere submanifolds in sphere 3. Real hypersurfaces in complex projective space 4. Austere hypersurfaces in S5 and real hypersurfaces inCP2 5. Austere hypersurfaces in S5 and Levi-flat real hypersurfaces in CP2 Acknowledgments References On the minimality of normal bundles in the tangent bundles over the complex space forms T. Kajigaya 1. Introduction 2. Preliminaries 2.1. Tangent bundles and the Sasaki metric 2.2. Lemmata for the general setting 3. On the minimality of normal bundles in the tangent bundle over the complex space form 3.1. Complex submanifolds 3.2. Totally geodesic submanifolds 3.3. Hopf hypersurfaces Acknowledgment References Over-determined systems on surfaces N. Ando 1. Introduction 2. Semisurfaces 3. The fundamental equations of surfaces 4. The compatibility condition 5. The existence of just two solutions 6. Over-determined systems on minimal surfaces in E3 7. A two-dimensional manifold equipped with a 1-form and two one-dimensional distributions 8. A generalization of an over-determined system on as urface References Author Index