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Submanifolds with nonpositive extrinsic curvature. (English) Zbl 1370.53043

The authors prove curvature estimates for complete isometric immersions \(f : M^m \rightarrow N^{n+ l} = P^n \times Q^l\) satisfying \(n+2l < 2m\) and which are cylindrically bounded, meaning that \(f(M) \subset B_P(R) \times Q\) where \(B_P(R)\) is an appropriately chosen geodesic ball of radius \(R\) in \(P\). The main tools used in the proof are Otsuki’s Lemma, the Omori-Yau maximum principle, and the Hessian comparison theorem.
This work generalizes a result by L. J. Alías et al. [Trans. Am. Math. Soc. 364, No. 7, 3513–3528 (2012; Zbl 1277.53047)].

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A35 Non-Euclidean differential geometry

Citations:

Zbl 1277.53047
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References:

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