A Willmore type inequality
Abstract
We show that a certain Willmore-type functional for closed submanifolds in a Cartan-Hadamard manifold has a lower bound which
depends on the dimension of the submanifold only.
We first recall some results which give lower bounds for certain energy functionals.
Theorem 1 (Fenchel [2, p. 399]). Suppose γ is a simple closed curve in R3
and k is its curvature, then
Z
|k| ≥ 2π.
γ
The equality holds if and only if γ is a plane convex curve.
There is a closely related to the following
Theorem 2 (Fary-Milnor [2, p. 402]). If γ is a knotted simple closed curve
in R3 with curvature k, then
Z
|k| ≥ 4π.
γ
Intuitively, γ is knotted means that we cannot deform γ continuously in R3
to the standard planar circle without crossing itself.
On the other hand, we have
Proposition 1. For a closed surface Σ in R3 , if H is is (un-normalized)
mean curvature, then
Z
H 2 ≥ 16π,
Σ
with the equality holds if and only if it is a round sphere.
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Proof. Let {λi }2i=1 be the principal curvatures. Then
H 2 − 4K = (λ1 + λ2 )2 − 4λ1 λ2 = (λ1 − λ2 )2 ≥ 0.
Let M+ = {K > 0}. Then
Z
Z
2
H ≥
M
Z
2
H ≥4
M+
K ≥ 16π.
M+
The last inequality is true because the Gauss map ν : M → S2 is surjective
onto the sphere which has area 4π and its Jacobian is exactly K.
Furthermore, recently Marques and Neves proved the following Willmore
conjecture:
Theorem 3 (Marques, Neves [4]). For any immersed torus Σ in R3 ,
Z
H 2 ≥ 8π 2 .
Σ
The equality holds if and only if Σ is the Clifford torus.
A very general result for submanifolds immersed in Rn is obtained by Chen
[1]. It is interesting to ask if a similar inequality holds for for a more general
ambient space. We note that a similar result holds when the ambient space
is a Cartan-Hadamard manifold:
Theorem 4. Let N n be a simply connected Riemannian manifold with nonpositive curvature (Cartan-Hadamard manifold). Let 2 ≤ m ≤ n be an
integer. Then there exists a constant C depending on m only such that for
any closed oriented smooth submanifold M m immersed in N , we have
Z
|H|m ≥ C,
M
where H is the mean curvature vector of M .
The following result is an instance of Theorem 2.1 in [3] (see also [5]) and
is crucial to the proof of Theorem 4:
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Theorem 5. Let N n be a simply connected Riemannian manifold with nonpositive curvature. Let 2 ≤ m ≤ n be integer. Then there exists a constant
Cm depending only on m such that for any oriented submanifold M m immersed in N and any compactly supported smooth function f on M , we have
m
≤ k∇f k1 + kf Hk1 .
Cm kf k m−1
(0.1)
Here H is the mean curvature vector of M and k · kp is the Lp norm on M .
The proof of Theorem 4 is now very straightforward.
Proof of Theorem 4. This follows immediately by choosing f = 1 in Theorem
5:
m−1
m−1
Cm Area(M ) m ≤ kHk1 ≤ kHkm Area(M ) m .
Remark 1.
1. When N = Rn , the optimal result is obtained by Chen
[1]. The optimal constant is exactly the area of the unit m-dimensional
sphere and the equality is attained when and only when M is an embedded sphere.
2. It is also interesting to ask if the constant in Theorem 4 can be improved
under various topological conditions on M .
References
[1] B. Chen. On a theorem of Fenchel-Forsuk-Willmore-Chern-Lashof.
Mathematische Annalen, 194(1):19–26, 1971.
[2] M.P. Do Carmo. Differential geometry of curves and surfaces, volume 2.
Prentice-Hall Englewood Cliffs, NJ:, 1976.
[3] D. Hoffman and J. Spruck. Sobolev and isoperimetric inequalities for
Riemannian submanifolds. Comm. on Pure and Appl. Math., 27(6):715–
727, 1974.
[4] F.C. Marques and A. Neves. Min-max theory and the Willmore conjecture. arXiv preprint arXiv:1202.6036, 2012.
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[5] J.H. Michael and L.M. Simon. Sobolev and mean-value inequalities
on generalized submanifolds of Rn . Comm. on Pure and Appl. Math.,
26(3):361–379, 1973.
[6] P. Topping. Relating diameter and mean curvature for submanifolds of
Euclidean space. Commentarii Mathematici Helvetici, 83(3):539–546,
2008.
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