Abstract
We derive the Simons-type equation for \(f\)-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed \(f\)-minimal hypersurfaces immersed in the product manifold \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with \(f=\frac{t^2}{4}\). Also, we classify closed \(f\)-minimal hypersurfaces with \(L_f\)-index one immersed in \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with the same \(f\) as above.
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1 Introduction
In [18], Simons proved an identity, called Simons’s equation, for the Laplacian of \(|A|^2\), the square of the norm of the second fundamental form of minimal hypersurfaces in Riemannian manifolds. Simons’s equation plays an important role in the study of minimal hypersurfaces. For self-shrinkers for the mean curvature flow in the Euclidean space \(\mathbb {R}^{n+1}\), the Simons-type equation also holds ([7, 10] Sect. 10.2). By applying it, Huisken [10] proved that an embedded closed self-shrinker of nonnegative mean curvature must be a sphere of radius \(\sqrt{2n}\), and recently Colding and Minicozzi [7] classified complete embedded self-shrinkers of nonnegative mean curvature with polynomial volume growth. Later, Le and Sesum [12] and Cao and Li [2] used it to obtain gap theorems for self-shrinkers.
Both minimal hypersurfaces and self-shrinkers are special cases of \(f\)-minimal hypersurfaces in the weighted Riemannian manifolds. See the definition of \(f\)-minimal hypersurfaces in Sect. 2 and more examples in [4]. An \(f\)-minimal hypersurface \(\varSigma \) is not only a critical point of the weighted volume functional \(\int _\varSigma e^{-f}d\sigma \) of \(\varSigma \), where \(d\sigma \) denotes the volume element of \((\varSigma , g)\), but also a minimal hypersurface in \((M, \tilde{g})\), where the new metric \(\tilde{g}=e^{-\frac{2}{n}f}\overline{g}\) of \(M\) is conformal to \(\overline{g}\). Recently, Lott [15] and Magni et al. [16] showed that \(f\)-minimal hypersurfaces arise in the study of the mean curvature flow of a hypersurface in an ambient manifold evolving by Ricci flow. Especially the mean curvature soliton (for the mean curvature flow of a hypersurface in a gradient Ricci soliton solution) introduced by Lott [15] is just an \(f\)-minimal hypersurface, where \(f\) is the potential function of the ambient gradient Ricci soliton.
Recently, Liu [14] studied stable \(f\)-minimal hypersurfaces in manifolds with nonnegative Bakry–Émery Ricci curvature and gave a partial classification of the ambient space when the dimension is \(3\) and \(f\) is bounded. The present authors [4] studied the stability condition and compactness of \(f\)-minimal surfaces. Li and Wei [13] gave eigenvalue estimates for closed \(f\)-minimal hypersurfaces in a compact manifold with positive \(m\)-Bakry–Émery curvature. We [3] also obtained similar eigenvalue estimates for some closed \(f\)-minimal hypersurfaces in a complete manifold with positive Bakry–Émery curvature. These estimates have been used to prove compactness theorems for closed \(f\)-minimal surfaces.
In this paper, we will prove a Simons-type equation for \(f\)-minimal hypersurfaces in a smooth metric measure space \((M, \overline{g}, e^{-f}d\mu )\), that is, an identity for the weighted Laplacian \(\Delta _f\) of \(|A|^2\) of \(f\)-minimal hypersurfaces, involving the Bakry–Émery Ricci curvature \(\overline{\mathrm{Ric}}_f\) (see Theorem 3). Also, we derive the equations for the weighted Laplacian \(\Delta _f\) of some other geometric quantities on \(f\)-minimal hypersurfaces, like the mean curvature \(H\), etc.; see, for instance, Propositions 1 and 2. Since these equations involve \(\overline{\mathrm{Ric}}_f\), we naturally would like to consider the cases in which the ambient manifolds are gradient Ricci solitons, that is, \(M\) satisfies \(\overline{\mathrm{Ric}}_f=C\overline{g}\); see Corollaries 2, 3 and 4. Further, we apply the equations mentioned above to the special case of \(f\)-minimal hypersurfaces in the cylinder shrinking soliton \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with \(f=\frac{t^2}{4}\), where \(t\) is the coordinate of the second factor \(\mathbb {R}\). Namely, we obtain the following pinching theorem.
Theorem 1
Let \(\varSigma ^{n}\) be a closed immersed \(f\)-minimal hypersurface in the product manifold \(\mathbb {S}^{n}\bigr (\sqrt{2(n-1)}\bigr )\times \mathbb {R}\), \(n\ge 3\), with \(f=\frac{t^2}{4}\), \(t\in \mathbb {R}\). If the square of the norm of the second fundamental form of \(\varSigma \) satisfies
then \(\varSigma \) is \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\), where \(\alpha =\langle \nu ,\frac{\partial }{\partial t}\rangle \), \(\nu \) is the outward unit normal to \(\varSigma \) and \(t\) denotes the coordinate of the factor \(\mathbb {R}\) of \(\mathbb {S}^{n}\bigr (\sqrt{2(n-1)}\bigr )\times \mathbb {R}\).
Observe that \(n\ge 3\) implies that \(\frac{8}{n-1}\alpha ^2(1-\alpha ^2)\le 1\) and hence the inequalities in Theorem 1 make sense. Theorem 1 implies that
Corollary 1
There is no closed immersed \(f\)-minimal hypersurface in the product manifold \(\mathbb {S}^{n}\bigr (\sqrt{2(n-1)}\bigr )\times \mathbb {R}\), \(n\ge 3\), with \(f=\frac{t^2}{4}\), \(t\in \mathbb {R}\) so that the square of its norm of the second fundamental form satisfies
Next, we discuss, as another application, the eigenvalues and the index of the operator \(L_f\) on \(f\)-minimal hypersurfaces. The eigenvalues of the \(L\)-operator for self-shrinkers were discussed in [7] and recently, Hussey [11] studied the index of the \(L\)-operator for self-shrinkers in \(\mathbb {R}^{n+1}\). Observe that the \(L\)-operator is just the \(L_f\) operator for self-shrinkers (see Example 1). In this paper, we classify closed \(f\)-minimal hypersurfaces in the cylinder shrinking soliton \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) whose \(L_f\) operators have index one and prove that
Theorem 2
Let \(\varSigma ^{n}\) be a closed immersed \(f\)-minimal hypersurface in the product manifold \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\) with \(f=\frac{t^2}{4}\). Then \(L_f\)-\(\hbox {ind}(\varSigma )\ge 1\). Moreover, the equality holds if and only if \(\varSigma =\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\).
For complete noncompact \(f\)-minimal hypersurfaces in the cylinder shrinking soliton \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) with \(f=\frac{t^2}{4}\), the first and third authors [6] proved the results corresponding to Theorems 1 and 2. In this case complete noncompact \(f\)-minimal hypersurfaces are assumed to have polynomial volume growth, which is equivalent to properness of immersion, or finiteness of weighted volume (see [5] and [4]).
The rest of this paper is organized as follows: In Sect. 2 some definitions and notation are given. In Sect. 3 we prove the Simons-type equation and the equations for \(\Delta _f\) of other geometric quantities. In Sect. 4, we calculate the index of the \(L_f\) operator on closed \(f\)-minimal hypersurfaces in the cylinder shrinking soliton \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\). In Sect. 5, we prove Theorem 1 for closed \(f\)-minimal hypersurfaces in the cylinder shrinking soliton \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\).
2 Definitions and Notation
Let \((M^{n+1}, \overline{g}, e^{-f}d\mu )\) be a smooth metric measure space, which is an \((n+1)\)-dimensional Riemannian manifold \((M^{n+1}, \overline{g})\) together with a weighted volume form \(e^{-f}d\mu \) on \(M\), where \(f\) is a smooth function on \(M\) and \(d\mu \) the volume element induced by the metric \(\overline{g}\). In this paper, unless otherwise specified, we denote by a bar all quantities on \((M, \overline{g})\), for instance by \(\overline{\nabla }\) and \(\overline{\mathrm{Ric}}\), the Levi-Civita connection and the Ricci curvature tensor of \((M, \overline{g})\) respectively. For \((M, \overline{g}, e^{-f}d\mu )\), the \(\infty \)-Bakry–Émery Ricci curvature tensor \(\overline{\mathrm{Ric}}_{f}\) (for simplicity, Bakry–Émery Ricci curvature), which is defined by
where \(\overline{\nabla }^{2}f\) is the Hessian of \(f\) on \(M\). If \(f\) is constant, \(\overline{\mathrm{Ric}}_{f}\) is the Ricci curvature \(\overline{\mathrm{Ric}}\).
Now let \(i: \varSigma ^{n}\rightarrow M^{n+1}\) be an \(n\)-dimensional smooth immersion. Then \(i\) induces a metric \(g=i^*\overline{g}\) on \(\varSigma \) so that \(i: (\varSigma ^{n}, g) \rightarrow (M, \overline{g})\) is an isometric immersion. We will denote for instance by \({\nabla }\), \(\mathrm{Ric}\), \(\Delta \) and \(d\sigma \), the Levi-Civita connection, the Ricci curvature tensor, the Laplacian, and the volume element of \((\varSigma , g)\) respectively.
The restriction of \(f\) on \(\varSigma \), still denoted by \(f\), yields a weighted measure \(e^{-f}d\sigma \) on \(\varSigma \) and hence an induced smooth metric measure space \((\varSigma ^{n}, g, e^{-f}d\sigma )\). The associated weighted Laplacian \({\Delta }_{f}\) on \(\varSigma \) is defined by
The second-order operator \({\Delta }_f\) is a self-adjoint operator on \(L^2(e^{-f}d\sigma )\), the space of square integrable functions on \(\varSigma \) with respect to the measure \(e^{-f}d\sigma \).
We define the second fundamental form \(A\) of \(\varSigma \) by
where \(p\in \varSigma , X, Y\in T_p \varSigma \), \(\nu \) is a unit normal vector at \(p\).
In a local orthonormal system \(\{e_i\}, i=1,\ldots , n\) of \(\varSigma \), the components of \(A\) are denoted by \(a_{ij}=A(e_i,e_j)=\langle \overline{\nabla }_{e_i}\nu , e_j\rangle \). The shape operator \(A\) and the mean curvature \(H\) of \(\varSigma \) are defined by
With the above notation, we have the following
Definition 1
The weighted mean curvature \(H_f\) of the hypersurface \(\varSigma \) is defined by
\(\varSigma \) is called an \(f\)-minimal hypersurface if it satisfies
Definition 2
The weighted volume of \(\varSigma \) is defined by
It is known that \(\varSigma \) is \(f\)-minimal if and only if it is a critical point of the weighted volume functional. On the other hand, we can view it in another manner: \(\varSigma \) being \(f\)-minimal in \((M,\overline{g})\) is equivalent to \((\varSigma , i^*\tilde{g})\) being minimal in \((M,\tilde{g})\), where the conformal metric \(\tilde{g}=e^{-\frac{2f}{n}}\overline{g}\) (cf. [4]).
Now we assume that \(\varSigma \) is a two-sided hypersurface, that is, there is a globally defined unit normal \(\nu \) on \(\varSigma \).
Definition 3
For a two-sided hypersurface \(\varSigma \), the \(L_f\) operator on \(\varSigma \) is given by
where \(|A|^2\) denotes the square of the norm of the second fundamental form \(A\) of \(\varSigma \).
The operator \(L_{f}=\Delta _{f}+|A|^2+\overline{\mathrm{Ric}}_{f}(\nu ,\nu )\) is called the \(L_{f}\)-stability operator of \(\varSigma \).
Example 1
For self-shrinkers in \(\mathbb {R}^{n+1}\), the operator \(L_f\), where \(f=\frac{|x|^2}{4}\), is just the \(L\) operator in [7]:
Definition 4
A two-sided \(f\)-minimal hypersurface \(\varSigma \) is said to be \(L_{f}\)-stable if for any compactly supported smooth function \(\varphi \in C_o^{\infty }(\varSigma )\), it holds that
or equivalently,
\(L_f\)-stability means that the second variation of the weighted volume of \(f\)-minimal hypersurface \(\varSigma \) is nonnegative. Further, one has the definition of the \(L_f\)-index of \(f\)-minimal hypersurfaces. Since \(\Delta _f\) is self-adjoint in the weighted space \(L^2(e^{-f}d\sigma )\), we may define a symmetric bilinear form on the space \(C_o^{\infty }(\varSigma )\) of compactly supported smooth functions on \(\varSigma \) by, for \(\varphi , \psi \in C_o^{\infty }(\varSigma )\),
Definition 5
The \(L_{f}\)-index of \(\varSigma \), denoted by \(L_{f}\)-\(\hbox {ind}(\varSigma )\), is defined to be the maximum of the dimensions of negative definite subspaces for \(B_f\).
In particular, \(\varSigma \) is \(L_f\)-stable if and only if \(L_{f}-\hbox {ind}(\varSigma )=0\). The \(L_f\)-index of \(\varSigma \) has the following equivalent definition: Consider the Dirichlet eigenvalue problems of \(L_f\) on a compact domain \(\Omega \subset \varSigma \):
\(L_{f}\)-\(\hbox {ind}(\varSigma )\) is defined to be the supremum over compact domains of \(\varSigma \) of the number of negative (Dirichlet) eigenvalues of \(L_f\) (cf. [9]).
It is known that an \(f\)-minimal hypersurface \((\varSigma ,{g})\) is \(L_f\)-stable if and only if \((\varSigma , i^*\tilde{g})\) is stable as a minimal surface in \((M,\tilde{g})\). Further, the Morse index of \(L_f\) on \((\varSigma ,g)\) is equal to the Morse index of the Jacobi operator on minimal hypersurface \((\varSigma , i^*\tilde{g})\) (see [4]).
We will take the following convention for tensors. For instance, under a local frame field on \(M\), suppose that \(T=(T_{j_1,\ldots , j_r})\) is an \((r,0)\)-tensor on \(M\). The components of the covariant derivative \(\overline{\nabla }T\) are denoted by \(T_{j_1,\ldots , j_r;i}\), that is,
Meanwhile, under a local frame field on \(\varSigma \), suppose that \(S=(S_{k_1,\ldots , k_s})\) is an \((s,0)\)-tensor on \(\varSigma \). The components of the covariant derivative \({\nabla }S\) are denoted by \(S_{k_1,\ldots , k_s,l}\), that is,
Throughout this paper, we assume that the \(f\)-minimal hypersurfaces are orientable and without boundary. For a closed hypersurface, we choose \(\nu \) to be the outer unit normal. Finally, we refer the interested reader to [1, 3, 4, 8] and the references therein for more details about \(f\)-minimal hypersurfaces.
3 Simons-Type Equation for \(f\)-Minimal Hypersurfaces
First, we calculate the weighted Laplacian \(\Delta _f\) for mean curvature \(H\) of \(f\)-minimal hypersurfaces.
Proposition 1
Let \((\varSigma ^{n}, g)\) be an \(f\)-minimal hypersurface isometrically immersed in a smooth metric measure space \((M,\overline{g},e^{-f}d\mu )\). Then the mean curvature \(H\) of \(\varSigma \) satisfies
or equivalently,
where \(\{e_1,\ldots , e_n\}\) is a local orthonormal frame field on \(\varSigma \), \(\nu \) denotes the unit normal to \(\varSigma \), and \(|_{\varSigma }\) denotes the restriction to \(\varSigma \).
Proof
We choose a local orthonormal frame field \(\{e_i\}_{i=1}^{n+1}\) for \(M\) so that, restricted to \(\varSigma \), \(\{e_i\}_{i=1}^{n}\) are tangent to \(\varSigma \), and \(e_{n+1}=\nu \) is the unit normal to \(\varSigma \). Throughout this paper, for simplicity of notation, we substitute \(\nu \) for the subscript \(n+1\) in the components of the tensors on \(M\), for instance, \(\overline{R}_{\nu ikj}=\overline{Rm}(\nu , e_i, e_k, e_j)\), \((\overline{\nabla }^2f)_{\nu i}=(\overline{\nabla }^2f)(\nu , e_i)\). Differentiating the mean curvature \(H=\langle \overline{\nabla }f,\nu \rangle \), we have, for \(1\le i\le n\),
Then for \(1\le i, j\le n\),
For a fixed point \(p\in \varSigma \), we may further choose the local orthonormal frame \(\{e_1,\ldots ,e_n\}\) so that \(\nabla _{e_i}e_j( p)=(\overline{\nabla }_{e_i}e_j)^{\top }(p)=0\), \(1\le i,j\le n\). Then at \(p\), for \(1\le i, j\le n\),
In the third equality of (10), we used the assumption: \(\nabla _{e_j}e_i( p)=0\), \(1\le i,j\le n\). Also by this assumption and the Codazzi equation, we have at \(p\), for \(1\le i,j\le n, \)
Substituting these equalities and (10) into (9), we have at \(p\), for \(1\le i,j\le n\),
On the other hand, it holds that on \(\varSigma \),
So
Substituting (12) into (11) and noting that \(f_{\nu }=H\), we have at \(p\), for \(1\le i,j\le n\),
Taking the trace, we have that at \(p\),
Since \(p\in \varSigma \) is arbitrary and (14) is independent of the choice of frame, (14) holds on \(\varSigma \). By (14) and \(\Delta _f=\Delta -\langle \nabla f,\nabla H\rangle \), we obtain (7) and also the equivalent identity (6).\(\square \)
Proposition 1 yields the following
Corollary 2
With the same assumption and notation as in Proposition 1,
or equivalently,
Next we will derive the Simons-type equation for \(f\)-minimal hypersurfaces.
Theorem 3
Let \((\varSigma ^{n},g)\) be an \(f\)-minimal hypersurface isometrically immersed in \((M,\overline{g},e^{-f}d\mu )\). Then the square of the norm of the second fundamental form of \(\varSigma \) satisfies
where the notation is the same as in Proposition 1.
Proof
Simons [18] proved the following identity (see, e.g., [17] (1.20)) under a local orthonormal frame \(e_1,\ldots , e_n\) of \(\varSigma \):
Observe that
Using the same local frame as in the proof of Proposition 1 and substituting (13), (19), (20), (21) and (22) into (18), we have that at \(p\), for \(1\le i,j\le n\),
Multiply (23) by \(a_{ij}\) and take the trace. Then it holds that at \(p\),
Thus
Since (25) is independent of the choice of the coordinates, (17) holds on \(\varSigma \).\(\square \)
When the ambient space \(M\) has the property \(\overline{\mathrm{Ric}}_f=C\overline{g}\), i.e., \(M\) is a gradient Ricci soliton, the Simons-type equation for \(f\)-minimal hypersurfaces is the following (26).
Corollary 3
Let \((M^{n+1},\overline{g}, e^{-f}d\mu )\) be a smooth metric measure space satisfying \(\overline{\mathrm{Ric}}_f=C\overline{g}\), where \(C\) is a constant. If \((\varSigma ,g)\) is an \(f\)-minimal hypersurface isometrically immersed in \(M\), then it holds that on \(\varSigma \)
where the notation is the same as in Theorem 3
Finally, in this section, we prove the following identity for \(L_f\) operator, which is useful for the study of the eigenvalues of \(L_f\) and \(L_f\)-index of \(f\)-minimal hypersurfaces.
Proposition 2
Let \((M^{n+1},\overline{g},e^{-f}d\mu )\) be a smooth metric measure space and \(X\) a parallel vector field on \(M\). If \((\varSigma ^{n},g)\) is an \(f\)-minimal hypersurface isometrically immersed in \(M\), then the function \(\alpha :\varSigma \rightarrow \mathbb {R}\) defined by \(\alpha =\langle X,\nu \rangle \) satisfies
where the notation is the same as in Theorem 3.
Proof
Choose a local field of orthonormal frame \(\{e_{1},\ldots ,e_{n},e_{n+1}\}\) on \(M\) as in the proof of Proposition 1. Then for \(1\le i\le n\),
Note that \(\nabla _{e_i}e_j(p)=0\), \(1\le i,j\le n\). The Hessian of \(\alpha \) at \(p\) is given by
Take the trace in (30). Then
Also, from (8) and (30) we have at \(p\),
Substituting the above identity into (31), we have
that is,
Since \(p\) is arbitrary and (33) is independent of the frame, we have proved (27) and then (28).\(\square \)
When the ambient manifold is a gradient Ricci soliton, we obtain
Corollary 4
Let \((M^{n+1},\overline{g}, e^{-f}d\mu )\) be a smooth metric measure space satisfying \(\overline{\mathrm{Ric}}_f=C\overline{g}\), where \(C\) is a constant. Suppose that \(X\) is a parallel vector field on \(\varSigma \). If \((\varSigma ,g)\) is an \(f\)-minimal hypersurface isometrically immersed in \(M\), then \(\alpha =\langle X,\nu \rangle \) satisfies that on \(\varSigma \),
Example 2
[7] Let \(M=\mathbb {R}^{n+1}\) and \(f=\frac{|x|^{2}}{4}\). The \(f\)-minimal hypersurfaces are self-shrinkers. Suppose that \(\varSigma \) is a self-shrinker. Then
If \(V\) is a constant vector in \(\mathbb {R}^{n+1}\) and \(\nu \) is the unit normal to \(\varSigma \), then
4 \(L_f\)-Index of \(f\)-Minimal Hypersurfaces
In this section, we study the \(L_{f}\)-index of closed \(f\)-minimal hypersurfaces immersed in the product manifold \(\mathbb {S}^{n}(a)\times \mathbb {R}\), \(n\ge 2\), with \(f(x,t)=\frac{(n-1)t^2}{4a^2}\), where \((x,t)\in \mathbb {S}^{n}(a)\times \mathbb {R}\) and \(\mathbb {S}^{n}(a)\) denotes the round sphere in \(\mathbb {R}^{n+1}\) of radius \(a\). For simplicity of notation, we only consider \(a=\sqrt{2(n-1)}\) and hence \(f=\frac{t^2}{4}\). The cases of other \(a\) are analogous. \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\) has the metric
where \(g_{\mathbb {S}^{n}(\sqrt{2(n-1)})}\) denotes the canonical metric of \(\mathbb {S}^{n}(\sqrt{2(n-1)})\). Let \(\{\overline{e}_{1},\ldots ,\overline{e}_{n+1}\}\) be a local orthonormal frame on \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\). By a straightforward computation, one has the components of the curvature tensor and Ricci curvature tensor of \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\) given by, for \(1\le i,j,k,l\le n+1\),
and
On the other hand,
Hence \((\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R},\overline{g},e^{-f}d\mu )\) is a smooth metric measure space with \(\overline{\mathrm{Ric}}_{f}=\frac{1}{2}\overline{ g}\). In addition, in the theory of Ricci flow, \((\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R},\overline{g},f)\) is a shrinking gradient soliton.
For an \(f\)-minimal hypersurface \(\varSigma \) immersed in \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\),
where \(\alpha =\langle \frac{\partial }{\partial t},\nu \rangle \). So \(\varSigma \) satisfies
The operator \(L_f\) on \(\varSigma \) is
Lemma 1
The slice \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\) is an \(f\)-minimal hypersurface in \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\). Moreover, a complete \(f\)-minimal hypersurface \(\varSigma \) is immersed in a horizontal slice \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{t\}\), where \(t\in \mathbb {R}\) is fixed, if and only if \(\varSigma \) is \( \mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\).
Proof
The unit normal \(\nu \) to \(\varSigma \) satisfies \(\nu =\frac{\partial }{\partial t}\) and hence \(AX=\overline{\nabla }_X\nu =0, X\in T\varSigma \). Thus \(\varSigma \) is totally geodesic. Meanwhile,
It follows that \(\varSigma \) is \(f\)-minimal if and only if \(t=0\). Further, by Gauss’s equation we know \(\varSigma \) has constant positive section curvature and hence is closed. Since the closed \(\varSigma \) has dimension \(n\) and \(\mathbb {S}^n\) is simply connected, \(\varSigma \) must be \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\).\(\square \)
We will prove that
Lemma 2
\(L_f\)-\(\hbox {ind}(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\})=1\).
Proof
On \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\), we have \(\nabla f=(\overline{\nabla }f)^{\top }=0\), \(|A|^{2}=0\). Hence,
Thus the eigenvalues of \(L_{f}\) are
where \(\lambda _{k}=\frac{k(k+n-1)}{2(n-1)}, k=0,1,\ldots \), are the eigenvalues of the Laplacian \(\Delta _{\mathbb {S}^{n}(\sqrt{2(n-1)})}\). Observe that
that is, \(L_{f}\) has a unique negative eigenvalue with multiplicity one. Therefore, the \(L_f\)-index of \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\) is \(1\).\(\square \)
We will prove Theorem 2, which says that \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\) is the unique closed \(f\)-minimal hypersurface of \(L_f\)-index one.
.
Proof of Theorem 2
On \(\varSigma \),
So
Since \(\varSigma \) is closed, there is a point \(p\in \varSigma \) such that \(t(p)=\max _{\varSigma } t\) and \(|\nabla t|(p)=0\). By equation (42), we have
Hence \(\alpha (p)=\pm 1\) and so \(\alpha \not \equiv 0\). Since \(\frac{\partial }{\partial t}\) is a parallel vector field on \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\) and \(\overline{\mathrm{Ric}}_{f}=\frac{1}{2}\overline{g}\), Proposition 2 implies that
Thus \(\alpha \) is an eigenfunction of \(L_f\) with eigenvalue \(-\frac{1}{2}\) and this implies that \(L_f\)-\(\mathrm ind (\varSigma )\ge 1\).
Now we consider the equality case. Lemma 2 says that \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\) has \(L_f\)-index one. Conversely, if \(L_f\)-\(\mathrm ind (\varSigma )=1\), then \(-\frac{1}{2}\) is the first eigenvalue. Then the corresponding eigenfunction \(\alpha \) cannot change sign. We may assume that \(\alpha >0\). On the other hand, \(L_{f}\alpha =\frac{1}{2}\alpha \) and \(\overline{\mathrm{Ric}}_f=\frac{1}{2}\overline{g}\) imply
Hence
By the maximum principle, \(\alpha \) is constant on \(\varSigma \). On the other hand, by (43), there is a point \(p\in \varSigma \) such that \(\alpha (p)=\pm 1\). Since \(\alpha \) is positive, \(\alpha \equiv 1\). Hence \(\nabla t=0\) and thus \(\varSigma \) is in a horizontal slice \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{t\}\). By Lemma 1, \(\varSigma \) must be \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\).\(\square \)
5 Pinching Theorem
First, we will derive various identities, including a Simons-type equation (see (47)) for \(f\)-minimal hypersurfaces immersed in \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\). Next, we apply them to obtain a pinching result for \(f\)-minimal hypersurfaces. We use the same notation as in Sect. 4.
Proposition 3
Let \(\varSigma \) be an \(f\)-minimal hypersurface immersed in the product manifold \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\) with \(f=\frac{t^{2}}{4}\). Then
Proof
Choose a local orthonormal frame field \(\{e_i\}_{i=1}^{n+1}\) for \(M\) so that, restricted to \(\varSigma \), \(\{e_i\}_{i=1}^{n}\) are tangent to \(\varSigma \), and \(e_{n+1}=\nu \) is the unit normal to \(\varSigma \). Recall that Proposition 2 states that
Substituting (38) in (48), we have
Now we prove (46). Note \(f=\frac{1}{4}t^2\). \(\overline{\nabla }^3 f=0\). This and Proposition 1 yield
Substituting (37) and (38) into (50), we have
Then,
In the above, we used \(H=\frac{t\alpha }{2}\) and \(\overline{\nabla }f=\frac{t}{2}\frac{\partial }{\partial t}\). Thus (46) holds. Finally we prove (47). Since \(\mathbb {S}^n(\sqrt{2(n-1)})\times \mathbb {R}\) is a symmetric space, \(\overline{\nabla }R=0\). By the Simons-type equation (Corollary 3), it holds that
Substituting the curvature tensors (35) into (51) and computing directly, we obtain
Note that the function \(\alpha \) satisfies \(\alpha _i=\displaystyle \sum _{j=1}^{n}a_{ij}\langle e_j,\frac{\partial }{\partial t}\rangle \). Hence,
Using \(\overline{\nabla }f=\frac{t}{2}\frac{\partial }{\partial t}\) and \(H=\frac{ t\alpha }{2}\), we obtain (47):
\(\square \)
Proposition 3 implies the following equations:
Lemma 3
If \(\varSigma \) is a closed orientable \(f\)-minimal hypersurface immersed in \(M=\mathbb {S}^{n}(\sqrt{2(n-1)})\times \mathbb {R}\), then
Proof
(52) can be obtained by integrating (45) directly. Now we prove (53). Since
Integrating (46) and using (55), we obtain
In the above we have used \(\int _\varSigma \Delta _fH^2e^{-f}=0\) and \(H^2=\alpha ^2f\). Thus, we get (53). Finally we prove (54). Integrating (47) and using (52) and (55), we have
\(\square \)
Using Lemma 3, we may prove Theorem 1.
.
Proof of Theorem 1
Observe that, for \(n\ge 3\),
if and only if
So (54) implies that on \(\varSigma \), for \(n\ge 3\),
and
Hence, \(|A|^{2}\) and \(H\) are constants. Substituting in (53), we obtain
So
This implies that on \(\varSigma \),
Since \(\varSigma \) is closed, \(\alpha ^2\equiv 1\). Without loss of generality, we choose \(\alpha \equiv 1\). So \(\varSigma \) is in a horizontal slice \(\mathbb {S}^n(\sqrt{2(n-1)})\times \{t\}\). By Lemma 1, we conclude that \(\varSigma \) is \(\mathbb {S}^{n}(\sqrt{2(n-1)})\times \{0\}\).\(\square \)
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Acknowledgments
The Xu Cheng and Detang Zhou are partially supported by CNPq and Faperj of Brazil. The Tito Mejia is supported by CNPq of Brazil.
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Communicated by Jiaping Wang.
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Cheng, X., Mejia, T. & Zhou, D. Simons-Type Equation for \(f\)-Minimal Hypersurfaces and Applications. J Geom Anal 25, 2667–2686 (2015). https://doi.org/10.1007/s12220-014-9530-1
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DOI: https://doi.org/10.1007/s12220-014-9530-1