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Finiteness of index and total scalar curvature for minimal hypersurfaces


Author: Johan Tysk
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
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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}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0946639-1
Keywords: Minimal hypersurface, index, eigenvalue, second fundamental form
Article copyright: © Copyright 1989 American Mathematical Society

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