In this post I will talk about a characterization of constant higher mean curvature hypersurfaces in space forms, the main result is a Schur-type lemma.

Let be a space form and let be a closed hypersurface. Let be a global unit normal field on , we define the second fundamental form by , where is the connection in .

We define the -th mean curvature of in as follows. Suppose are the principal curvatures of (eigenvalues of ), we define by the identity

We define the -th Newton transformation of (which can be regarded as a tensor) for as the self-adjoint linear map

Alternatively, they can be defined inductively by

Clearly, and so they can be simultaneously diagonalized, i.e. has the same eigenvectors as .

In a local frame, the matrix entries of are given by ([Reilly] Proposition A)

where are the matrix entries of and are the generalized Kronecker delta symbols. This can be seen by comparing the two sides using a local frame which is a basis consisting of the eigenvectors of .

Lemma 1Let and be the connection and trace on respectively, then for all ,

*Proof:*

The first equation follows from

where we have used the Codazzi equation in the third line (as has constant curvature). The second assertion follows by applying the Newton’s identities (hence the name Newton tensors) to (1), where .

Theorem 2Let be a closed hypersurface in a space form, be the average of and be the traceless part of . Suppose the Ricci curvature of is non-negative, then for , we have

*Proof:*

We can assume . Let be the solution to

The solution exists because . As and ,

Thus (5) becomes

given . Substitute this into (4), we have

As , we have

Thus (6) can be rephrased as

Remark 1When , as , it is easy to see that where is the traceless part of the second fundamental form. Note also that is the mean curvature. Thus (3) can be rewritten as

Remark 2For a hypersurface in Euclidean space, having nonnegative Ricci curvature is equivalent to (i.e. convex), see \cite{Perez} p. 48. So the curvature assumptions in Theorem 2 can be replaced by being convex.

Theorem 2 immediately implies the following Schur-type lemma:

Corollary 3With the same assumptions as in Theorem 2, if , then is constant.

By the way, this is called a Schur-type lemma because in [Lellis-Topping], it was proved in a similar way that if ()is a closed Riemannian manifold with nonnegative Ricci curvature, then

Clearly this implies that under the stated assumption, the Schur’s lemma holds: if , i.e. is Einstein (), then is constant (the original version has no restriction on and for being closed).

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