A Willmore type inequality
Abstract
We show that a certain Willmore-type functional for closed sub- manifolds 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 func- tionals.
Theorem 1 (Fenchel [2, p. 399]). Suppose γ is a simple closed curve in R3 and k is its curvature, then
Z
γ
|k| ≥ 2π.
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
γ
|k| ≥ 4π.
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
Σ
H2 ≥ 16π,
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
H2− 4K = (λ1+ λ2)2− 4λ1λ2 = (λ1− λ2)2 ≥ 0.
Let M+= {K > 0}. Then Z
M
H2 ≥ Z
M+
H2 ≥ 4 Z
M+
K ≥ 16π.
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
Σ
H2 ≥ 8π2.
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 Nn be a simply connected Riemannian manifold with non- positive 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 Mm immersed in N , we have
Z
M
|H|m ≥ C,
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 Nn be a simply connected Riemannian manifold with non- positive curvature. Let 2 ≤ m ≤ n be integer. Then there exists a constant Cm depending only on m such that for any oriented submanifold Mm im- mersed in N and any compactly supported smooth function f on M , we have
Cmkf k m
m−1 ≤ k∇f k1 + kf Hk1. (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:
CmArea(M )m−1m ≤ kHk1 ≤ kHkmArea(M )m−1m .
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 em- bedded 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 conjec- ture. 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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