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Miscellany Results for Submanifolds

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Maximum Principles and Geometric Applications

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Abstract

This chapter is basically devoted to miscellany applications of the results presented in Chaps. 3 and 4. We show how, with the aid of various forms of the maximum principle, we can improve on some classical results. In fact we begin with some introductory considerations to motivate a nonimmersibility result for a manifold M into cones of the Euclidean space due to Omori [210] and of which we provide an improved version in Theorem 5.1. We recall that, in the cited work of Omori, we have the first appearance of what is now known in the literature as the Omori-Yau maximum principle. We then continue our investigation by establishing a quantitative estimate, according to the results presented in [181], for the width of the cone of \(\mathbb{R}^{n}\) containing the image of M under a smooth map, see Theorem 5.2. Later, we elaborate on some old result of Jorge and Koutroufiotis [154], (see Theorem 5.6) and we provide a “quantitative” version for immersions into a cone with the aid of the WMP for the Hessian (see Theorem 5.7). With the help of this result and of the theory of flat bilinear forms we are able to consider also the case where M is a Kähler manifold in Corollary 5.7.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain

    Luis J. Alías

  2. Dipartimento di Matematica, Università degli Studi di Milano, Milan, Italy

    Paolo Mastrolia & Marco Rigoli

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  1. Luis J. Alías
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  2. Paolo Mastrolia
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  3. Marco Rigoli
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Alías, L.J., Mastrolia, P., Rigoli, M. (2016). Miscellany Results for Submanifolds. In: Maximum Principles and Geometric Applications. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-24337-5_5

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