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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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