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Maximum principles and nonexistence results for minimal submanifolds

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Abstract

We construct a quadratic form on ℝn+k of signature (n-k) which is subharmonic on any n-dimensional minimal submanifold in ℝn+k. This yields an improvement over the convex hull property of minimal submanifolds as well as necessary conditions for compact minimal submanifolds the boundaries of which lie in disconnected sets. The argument also extends to submanifolds of bounded mean curvature. Furthermore an optimal nonexistence result is derived by employing a different geometrical argument, which is based on the construction of n-dimensional catenoids.

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  1. Fachbereich Mathematik, Universität des Saarlandes, D-6600, Saarbrücken

    Ulrich Dierkes

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  1. Ulrich Dierkes
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Dierkes, U. Maximum principles and nonexistence results for minimal submanifolds. Manuscripta Math 69, 203–218 (1990). https://doi.org/10.1007/BF02567919

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  • DOI: https://doi.org/10.1007/BF02567919

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