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On minimal hypersurfaces with finite harmonic indices. (English) Zbl 1023.53046
In this interesting paper, the authors introduce the concept of harmonic index and harmonic stability for complete, minimal hypersurfaces $$M$$ in $$\mathbb{R}^{n+1}$$, $$n\geq 3$$. It is shown that a stable complete, minimal hypersurface in $$\mathbb{R}^{n+1}$$ is also harmonically stable.
Various results are proved. For instance, if $$e(M)$$ denotes the number of ends of $$M$$ and $$h(M)$$ is the harmonic index of $$M$$, then $$e(M)=h(M)+1$$ and $h(M)\leq c(n)\int_M|A|^n dM,$ where $$c(n)$$ is a constant dependent only on $$n$$ and $$|A|$$ is the norm of the second fundamental form of $$M$$. If the integral of the second member of the above inequality is finite, $$M$$ is said to have finite total curvature. It is proved that the only orientable complete harmonic stable hypersurfaces with finite total curvature are hyperplanes.
To each end $$E_i$$ in $$M$$, the authors associate a nonnegative harmonic function $$u_i$$. These functions span an $$e(M)$$-dimensional vector space and form a partition of unity of $$M$$. Furthermore, if $$V$$ is the space of bounded harmonic functions on a complete minimal hypersurface $$M$$ with finite total curvature, then $$\dim V=e(M)$$ and $$V=\text{span}\{u_1,u_2,\dots, u_{e(M)}\}$$.

MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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