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Constant higher-order mean curvature hypersurfaces in Riemannian spaces. (English) Zbl 1118.53038
The authors consider the following question: We start by fixing a configuration. All manifolds are connected and orientable; superscripts denote dimensions and are dropped when clear from the context. Let $$\overline M^{n+1}$$ be a Riemannian manifold. Fix a totally geodesic hypersurface $$P^n\subset\overline M^{n+1}$$ and let $$\Sigma^{n-1}\subset P$$ be a compact embedded submanifold in $$P^n$$. Let $$M^n$$ be a manifold with boundary $$\partial M$$. $$M$$ is said to be a hypersurface of $$\overline M$$ with boundary $$\Sigma$$ if there exists an immersion $$\psi: M^n\to\overline M^{n+1}$$ such that $$\psi$$ restricted to $$\partial M$$ is a diffeomorphism onto $$\Sigma$$.
The question referred to above is as follows: How is the geometry of $$M$$ related to the geometry of the inclusion $$\Sigma\subset P$$?
A first answer is given by a series of expressions of which the simplest one is $\langle T_r \nu,\nu \rangle=(-1)^r s_r\langle\xi,\nu\rangle^r,\quad 1\leq r \leq n-1.$ Here $$T_r$$ is the $$r$$th Newton tensor associated to the second fundamental form of $$M$$, $$\nu$$ is the outward unit conormal vector field along $$\partial M$$, $$\xi$$ is the unit normal of $$P\subset\overline M$$, and $$s_r$$ are the elementary symmetric functions of the principal curvatures of $$\Sigma\subset P$$.
A geometric consequence of the above is the following theorem that generalizes an earlier result of Alias and Malacarne for $$=2$$.
Theorem 1. Let $$\Sigma^{n-1}$$ be a strictly convex compact submanifold of a hyperplane $$P^n\subset\mathbb R^{n+1}$$ and let $$\psi: M^n\to\mathbb R^{n+1}$$ be a compact embedded hypersurface with boundary $$\Sigma$$. Assume that the $$r$$th mean curvature $$H_r$$ of $$M$$ is a nonzero constant, $$2\leq r\leq n$$. Then $$M$$ inherits all symmetries of $$\Sigma$$. In particular, if $$\Sigma^{n-1}$$ is a round sphere, then $$M$$ is a spherical cap.
This solves a classical problem for $$r\neq 1$$. As far as we know the case $$r=1$$ is still open. A short survey of this problem is found in the paper.
One of the goals of the present paper is to generalize Theorem 1 to the hyperbolic space and to the sphere. For that, it is convenient to have a flux formula which the authors derive in the general case where the ambient space is a Riemannian manifold. The simplest case of this formula is when we are considering the ambient space as a space form, and it reads as follows:
Theorem 2. Let $$\psi: M^n\to \overline M^{n+1}$$ be an immersed, compact, oriented hypersurface with boundary $$\partial M$$, and Let $$D^n$$ be a compact imersed hypersurface with boundary $$\partial D=\partial M$$. Assume that $$M\cup D$$ is an oriented $$n$$-cycle of $$\overline M$$ and let $$N$$ and $$n_D$$ be the unit normal fields that orient $$M$$ and $$D$$, respectively. Assume further that $$\overline M$$ has constant sectional curvature and that the $$r$$th mean curvature $$H_r$$ of $$M$$ is constant, $$1\leq r\leq n$$. Then, for every Killing vector field, we have the following flux formula: $\int_{\partial M} \langle T_{r-1}\nu,Y\rangle\, ds=- r\binom{n}{r} H_r\int_D \langle Y, n_D\rangle\, dD.$
This formula has very interesting applications that we will not describe here for lack of space. However, we will describe its application to generalize Theorem 1 to the hyperbolic space $$\mathbb H^{n+1}$$ and the sphere $$S^{n+1}$$.
Theorem 1’. Let $$\Sigma^{n-1}$$ be a strictly convex compact submanifold of a totally geodesic hyperplane $$P^n\subset \mathbb H^{n+1}$$, and let $$M^n\subset\mathbb H^{n+1}$$ be a compact embedded hypersurface with boundary $$\Sigma$$. Assume that the $$r$$th mean curvature $$H_r$$ of $$M$$, $$2\leq r\leq n$$, is a nonzero constant. Then $$M$$ has all the symmetries of $$\Sigma$$; in particular, when $$\Sigma$$ is a geodesic sphere in $$P^n$$, $$M$$ is a spherical cap.
The spherical case is similar with the additional assumption that $$M$$ is contained in an open hemisphere $$S^{n+1}_+$$.

##### 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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