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Non-holomorphic Kaehler submanifolds of Euclidean space

Sergio Chion, Marcos Dajczer

Abstract

This paper is about non-holomorphic isometric immersions of Kaehler manifolds into Euclidean space $f\colon M^{2n}\to\R^{2n+p}$, $p\leq n-1$, with low codimension $p\leq 11$. In particular, it addresses a conjecture proposed by J. Yan and F. Zheng. The claim that if the index of complex relative nullity of the submanifold satisfies $ν_f^c<2n-2p$ at any point, then $f(M)$ can be realized as a holomorphic submanifold of a non-holomorphic Kaehler submanifold of $\R^{2n+p}$ of larger dimension and some large index of complex relative nullity. This conjecture had previously been confirmed by Dajczer-Gromoll for codimension $p=3$, and then by Yan-Zheng for $p=4$. For codimension $p\leq 11$, we already showed that the pointwise structure of the second fundamental form of the submanifold aligns with the anticipated characteristics, assuming the validity of the conjecture. In this paper, we confirm the conjecture until codimension $p=6$, whereas for codimensions $7\leq p\leq 9$ it is also possible that the submanifold exhibits a complex ruled structure with rulings of a specific dimension. Moreover, we prove that the claim of the conjecture holds for codimensions $7\leq p\leq 11$ albeit subject to an additional assumption.

Non-holomorphic Kaehler submanifolds of Euclidean space

Abstract

This paper is about non-holomorphic isometric immersions of Kaehler manifolds into Euclidean space , , with low codimension . In particular, it addresses a conjecture proposed by J. Yan and F. Zheng. The claim that if the index of complex relative nullity of the submanifold satisfies at any point, then can be realized as a holomorphic submanifold of a non-holomorphic Kaehler submanifold of of larger dimension and some large index of complex relative nullity. This conjecture had previously been confirmed by Dajczer-Gromoll for codimension , and then by Yan-Zheng for . For codimension , we already showed that the pointwise structure of the second fundamental form of the submanifold aligns with the anticipated characteristics, assuming the validity of the conjecture. In this paper, we confirm the conjecture until codimension , whereas for codimensions it is also possible that the submanifold exhibits a complex ruled structure with rulings of a specific dimension. Moreover, we prove that the claim of the conjecture holds for codimensions albeit subject to an additional assumption.
Paper Structure (8 sections, 27 theorems, 170 equations)

This paper contains 8 sections, 27 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex ruled Kaehler submanifolds
  4. Kaehler extension
  5. Complex ruled submanifold
  6. A certain vector subbundle
  7. Some estimates
  8. The proofs

Key Result

Theorem 1

. Let $f\colon M^{2n}\to\mathbb{R}^{2n+p}$, $p\leq n-1$ and $3\leq p\leq 9$, be a real Kaehler submanifold with $\nu_f^c(x)<2n-2p$ at any $x\in M^{2n}$. Assume for $7\leq p\leq 9$ that $f$ restricted to any open subset of $M^{2n}$ is not $(2n-t)$-complex ruled for $(p,t)=(7,8),(8,12)$ or $(9,16)$.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Proposition 7
  • Theorem 8
  • Proposition 9
  • Theorem 10
  • ...and 17 more