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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Finiteness of index and total scalar curvature for minimal hypersurfaces
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by Johan Tysk
Proc. Amer. Math. Soc. 105 (1989), 429-435
DOI: https://doi.org/10.1090/S0002-9939-1989-0946639-1

Abstract:

Let ${M^n},n \geq 3$, be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for $n = 3,4,5,{\text { or }}6$, the index of ${M^n}$ is finite if and only if the total scalar curvature of ${M^n}$ is finite, provided that the volume growth of ${M^n}$ is bounded by a constant times ${r^n}$, where $r$ is the Euclidean distance function. We also note that this result does not hold for $n \geq 8$. Moreover, we show that the index of ${M^n}$ is bounded by a constant multiple of the total scalar curvature for all $n \geq 3$, without any assumptions on the volume growth of ${M^n}$.
References
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Bibliographic Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 105 (1989), 429-435
  • MSC: Primary 53C42; Secondary 58C40, 58E15
  • DOI: https://doi.org/10.1090/S0002-9939-1989-0946639-1
  • MathSciNet review: 946639