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Two-ended hypersurfaces with zero scalar curvature. (English) Zbl 0929.53033

In many ways hypersurfaces with null $r$-th curvature function ${H}_{r}$ behave much like the minimal ones $\left({H}_{1}=0\right)$. One such manifestation is the following result proved in this paper, which extends to scalar-flat hypersurfaces $\left({H}_{2}=0\right)$ a well-known theorem of R. Schoen.

Theorem. The only complete scalar-flat embeddings ${M}^{n}\subset {ℝ}^{n+1}$, free of flat points, which are regular at infinity and have two ends, are the hypersurfaces of revolution.

The main tools in the proof of this and other related results presented in this paper are the maximum principle for the nonlinear equation ${H}_{r}\left(\text{graph}u\right)=0$ and the property that any height function $h$ of a hypersurface with null ${H}_{r}$ satisfies the intrinsic linear equation $\text{div}\left({T}_{r-1}\nabla h\right)=0$, where ${T}_{r-1}$ denotes the Newton tensor of the hypersurface shape operator.

##### MSC:
 53C42 Immersions (differential geometry) 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)