## Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space.(English)Zbl 1038.53060

The authors prove the following Theorem 3. Let $$\Sigma$$ be a strictly convex compact $$(n-1)$$-dimensional submanifold contained in a hyperplane $$\Pi\subset{\mathbb R}^{n+1}$$, and let $$\psi:M^n\to{\mathbb R}^{n+1}$$ be an embedded compact hypersurface with boundary $$\Sigma$$. Let us assume that, for a given $$2\leq r\leq n$$, the $$r$$-mean curvature $$H_r$$ of $$M$$ is a non-zero constant. Then $$M$$ is contained in one of the half-spaces of $${\mathbb R}^{n+1}$$ determined by $$\Pi$$ and $$M$$ has all the symmetries of $$\Sigma$$.
As a consequence, when a hypersurface has spherical boundary $$\Sigma$$ and constant $$r$$th mean curvature $$H_r$$ ($$r\geq2$$), then it is a hyperplanar round ball or a spherical cap (Cor. 4) and, in particular, this is the case when the scalar curvature, $$r=n$$, is constant (Theorem 1).

### MSC:

 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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