Finiteness of index and total scalar curvature for minimal hypersurfaces

Abstract

Let M n , n ≥ 3 {M^n},n \geq 3 , be an oriented minimally immersed complete hypersurface in Euclidean space. We show that for n = 3 , 4 , 5 , or 6 n = 3,4,5,{\text { or }}6 , the index of M n {M^n} is finite if and only if the total scalar curvature of M n {M^n} is finite, provided that the volume growth of M n {M^n} is bounded by a constant times r n {r^n} , where r r is the Euclidean distance function. We also note that this result does not hold for n ≥ 8 n \geq 8 . Moreover, we show that the index of M n {M^n} is bounded by a constant multiple of the total scalar curvature for all n ≥ 3 n \geq 3 , without any assumptions on the volume growth of M n {M^n} .