## 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
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## 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