The second variational formula for Willmore submanifolds

which is a conformal invariant under Moebius (or conformal) transformations of S n (see [3],
[17], [19]), where S is the square of the length of the second fundamental form, H is the mean
curvature vector and dM is the volume element of the induced metric on M. In case of a
surface in S 3 , this functional is equivalent to the Willmore functional We(M) := 4 �
M H2 dM
for surfaces in R 3 . The so-called Willmore conjecture states that We(M) ≥ 8π 2 for any
embedded torus in R 3 , well-studied (cf. [2], [9], [19], [20], [21]) since 1965. An equivalent
version of the Willmore conjecture states that W (T 2 ) ≥ 8π 2 for any embedded torus T 2 in
S 3 and the equality holds if and only if T 2 is Moebius equivalent to the Clifford torus. Many
efforts and partial results have been obtained for this conjecture, but up to now it is still
open.
From the analytic point of view, the second variational formula of the Willmore functional
might be very important to solve the Willmore conjecture. In 1978, J. L. Weiner gave the
second variational formula for minimal surfaces, which is a special class of Willmore surfaces
(cf. [18]). But it is rather complicated to give the second variation formula for general
Willmore surfaces by using Euclidean invariants. In [12] M. Peterson derives such a formula.
In [10] and [11] B. Palmer calculated the second variation of a Willmore surface in S 3 by
using its conformal Gauss map and used the formula to study the nonexistence of stable
Willmore surfaces under some conditions.
Since the Willmore functional (0.1) is invariant under the Moebius group (cf. [3], [17]),
one can use the framework of Moebius geometry and Moebius invariants to calculate the
second variational formula. It is the key point of this paper. For any submanifold M in
S n we can introduce a Moebius invariant metric g on M. Then the Willmore functional
is exactly the volume functional of g. The third author computed the first variation and
got the Euler - Lagrange equations in [17]. Submanifolds in S n satisfying these equations
are called Willmore submanifolds or Moebius minimal submanifolds. In this paper we give
the second variational formula of the Willmore functional for submanifolds in S n by using
Moebius invariants. Although this formula looks very complicated, in case of surfaces in
S 3 (which is the most important case) the formula is in good form (cf. §2, (2.43)). Using
the Euler-Lagrange � equations we find the standard examples of Willmore hypersurfaces
(m − k)/m) × Sm−k �
( k/m), 1 ≤ k ≤ m − 1} in Sm+1 , which is (euclidean)
{W m k := Sk (
minimal if and only if 2k = m. In some sense, W m � k can be considered � as the dual hypersurface
of the standard minimal hypersurface S k (
k/m) × S m−k (
that the hypersurface W m k are stable Willmore hypersurfaces.
(m − k)/m) in S m+1 . We show
We organize this paper as follows. In §1 we give Moebius invariants and local formulas in
Moebius geometry for submanifolds in S n . In §2 we calculate the second variation formula
for Willmore submanifolds in S n . As an application we prove in §3 that {W m k } are stable
Willmore hypersurfaces.
§1. Moebius invariants and local formulas for submanifolds in S n
Let x0 : M → S n be an m-dimensional compact submanifold with boundary ∂M, {e1, ..., em}
be a local orthonormal basis of T M with respect to the induced metric dx0 · dx0 and
{θ1, ..., θm} be its dual basis. Let {em+1, ..., en} be the local normal orthonormal vector
field. We make use of the following convention on the ranges of indices:
1 ≤ i, j, k, · · · ≤ m; m + 1 ≤ α, β, γ, . . . ≤ n
2