Real Kähler submanifolds in codimension 6

Carvalho, Alcides de; Guimarães, Felippe

doi:10.1090/proc/14737

Cited by 4 publications

(9 citation statements)

References 9 publications

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“…Remark 18. The proof of Theorem 1 in [1] makes use of Theorem 1 in [6] but that result does not hold for p = 6 as was clarified in [7]. Nevertheless, it was also established in [7] that Theorem 1 in [6] still holds for p = 6 under a slightly stronger assumption which happens to be satisfied in our case of real Kaehler submanifolds.…”

Section: Proof Of Theoremmentioning

confidence: 90%

“…Moreover, we have that our proof fails for p = 12, indicating that proposing p = 11 in the conjecture as the largest codimension seems appropriate. For codimension p ≤ 6 our result was obtained in [1] up to some inconsistencies in the argument; see Remark 18. As for higher codimension, it is shown by this paper that the proof is much more difficult.…”

mentioning

confidence: 87%

“…A bold conjecture by Yan and Zheng in [16] states, under the same assumption as above on the index of complex relative nullity, that any real Kaehler submanifold in codimension p ≤ 11 is a composition as in (1) along connected components of an open dense subset of M 2n . The purpose of this paper is to walk a fundamental step in order to treat that rather challenging conjecture.…”

mentioning

confidence: 99%

“…It is clearly relevant to establish conditions asserting that a real Kaehler submanifold is locally a composition as in (1). This was achieved for p = 3 by Dajczer and Gromoll [10] and for p = 4 by Yan and Zheng [16] under the assumption that the index of complex relative nullity of f : M 2n → R 2n+p satisfies ν c f (x) < 2n − 2p at any x ∈ M 2n .…”

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confidence: 99%

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The second fundamental form of the real Kaehler submanifolds

Chión¹,

Dajczer

2023

Preprint

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Let $f\colon M^{2n}\to\R^{2n+p}$, $2\leq p\leq n-1$, be an isometric immersion of a Kaehler manifold into Euclidean space. Yan and Zheng conjectured in \cite{YZ} that if the codimension is $p\leq 11$ then, along any connected component of an open dense subset of $M^{2n}$, the submanifold is as follows: it is either foliated by holomorphic submanifolds of dimension at least $2n-2p$ with tangent spaces inthe kernel of the second fundamental form whose images are open subsets of affine vector subspaces, or it is embedded holomorphically in a Kaehler submanifold of $\R^{2n+p}$ of larger dimension than $2n$. This bold conjecture was proved by Dajczer and Gromoll just for codimension three and then by Yan and Zheng for codimension four. In this paper we prove that the second fundamental form of the submanifold behaves pointwise as expected in case that the conjecture is true. This result is a first fundamental step fora possible classification of the non-holomorphic Kaehler submanifolds lying with low codimension in Euclidean space. A counterexample shows that our proof does not work for higher codimension, indicating that proposing $p=11$ in the conjecture as the largest codimension is appropriate.

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Section: Proof Of Theoremmentioning

confidence: 90%

mentioning

confidence: 87%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The second fundamental form of the real Kaehler submanifolds

Chión¹,

Dajczer

2023

Preprint

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“…Of course, we may be in the situation where f itself is holomorphic. Finally, the case of codimension p = 6 was considered by Carvalho and Guimarães [2].…”

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confidence: 99%

Holomorphicity of real Kaehler submanifolds

Carvalho¹,

Chion²,

Dajczer³

2019

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Let f : M 2n → R 2n+p denote an isometric immersion of a Kaehler manifold of complex dimension n ≥ 2 into Euclidean space with codimension p. If 2p ≤ 2n − 1, we show that generic rank conditions on the second fundamental form of the submanifold imply that f has to be a minimal submanifold. In fact, for codimension p ≤ 11 we prove that f must be holomorphic with respect to some complex structure in the ambient space.

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Minimal real Kaehler submanifolds

Chion¹,

Dajczer²

2022

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We show that generic rank conditions on the second fundamental form of an isometric immersion f : M 2n → R 2n+p of a Kaehler manifold of complex dimension n ≥ 2 into Euclidean space with low codimension p imply that the submanifold has to be minimal. If M 2n if simply connected, this amounts to the existence of a one-parameter associated family of isometric minimal immersions unless f is holomorphic.

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Real Kähler submanifolds in codimension 6

Cited by 4 publications

References 9 publications

The second fundamental form of the real Kaehler submanifolds

The second fundamental form of the real Kaehler submanifolds

Holomorphicity of real Kaehler submanifolds

Minimal real Kaehler submanifolds

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