# Minimal submanifolds immersed in a complex projective space

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## Abstract

Minimal submanifolds immersed in a complex projective space By Mayuko Kon SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE AT HOKKAIDO UNIVERSITY WEST 8, NORTH 10, KITA-KU, SAPPORO, 060-0810, JAPAN MARCH 2008 Contents 1 Introduction 2 2 Preliminaries 6 3 Laplacian 17 4 Integral formulas 24 5 Pinching theorems of the square of the length of the second fundamental form 29 6 Semi-flat normal connection 36 7 Pinching problem of the sectional curvature 40 8 Reduction of the codimension 46 9 Pinching theorems of the Ricci curvature 56 10 Real hypersurfaces of a complex space form 62 11 References 73 1 1 Introduction In 1968, Simons [34] gave the formula for the square of the length of the second fundamental form A of a compact n-dimensional minimal submani- fold M in a real space form Mn+p(k) of constant curvature k. The specific expression of the formula is the following: 1 2 ∆|A|2 = nk|A|2 −∑ a,b (trAvaAvb) 2 + ∑ a,b tr[Ava , Avb ] 2 + |∇A|2, where {v1, · · · , vp} is an orthonormal basis of the normal vector space. Here we denote by | · | the length of a tensor with respect to the induced metric g on M and by [ , ] the commutator. As an application, Simons proved that if the second fundamental form A of a compact n-dimensional minimal submanifold M in Sn+p satisfies |A|2 < n/(2 − 1/p), then M is totally geodesic. Moreover, Chern, do Carmo and Kobayashi [7] proved that if the second fundamental form A satisfies |A|2 = n/(2 − 1/p), then M is the Clifford hypersurface or the Veronese surface in S4. For minimal submanifolds in the sphere, the Simons type formula was studied by many authors, and many interesting results are given (e.g. [31], [42], [40]). For minimal submanifolds of complex space forms, there are some pinch- ing theorems with respect to the sectional curvature, Ricci curvature, scalar curvature and so on. For example, for the study of complex submanifolds in a complex spac

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