Constant higher-order mean curvature hypersurfaces in Riemannian spaces.

*(English)*Zbl 1118.53038The 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 \(S^{n+1}_+\).

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 |