Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 4, Issue 4, Article 74, 2003
THE SPREAD OF THE SHAPE OPERATOR AS CONFORMAL INVARIANT ˘ BOGDAN D. SUCEAVA D EPARTMENT OF M ATHEMATICS C ALIFORNIA S TATE U NIVERSITY, F ULLERTON CA 92834-6850, U.S.A.
[email protected] URL: http://math.fullerton.edu/bsuceava
Received 01 May, 2003; accepted 27 September, 2003 Communicated by S.S. Dragomir
A BSTRACT. The notion the spread of a matrix was first introduced fifty years ago in algebra. In this article, we define the spread of the shape operator by applying the same idea to submanifolds of Riemannian manifolds. We prove that the spread of shape operator is a conformal invariant for any submanifold in a Riemannian manifold. Then, we prove that, for a compact submanifold of a Riemannian manifold, the spread of the shape operator is bounded above by a geometric quantity proportional to the Willmore-Chen functional. For a complete non-compact submanifold, we establish a relationship between the spread of the shape operator and the Willmore-Chen functional. In the last section, we obtain a necessary and sufficient condition for a surface of rotation to have finite integral of the spread of the shape operator. Key words and phrases: Principal curvatures, Shape operator, Extrinsic scalar curvature, Surfaces of rotation. 2000 Mathematics Subject Classification. 53B25, 53B20, 53A30.
1. I NTRODUCTION In the classic matrix theory spread of a matrix has been defined by Mirsky in [7] and then mentioned in various references, as for example [6]. Let A ∈ Mn (C), n ≥ 3, and let λ1 , . . . , λn be the characteristic roots of A. The spread of A is defined to be s(A) = maxi,j |λi −λj |. Let us Pm,n 2 denote by ||A|| the Euclidean norm of the matrix A, i.e.: ||A|| = i,j=1 |aij |2 . We use also the classical notation E2 for the sum of all 2-square principal subdeterminants of A. If A ∈ Mn (C) then we have the following inequalities (see [6]): 12 2 (1.1) s(A) ≤ 2||A||2 − |trA|2 , n (1.2) ISSN (electronic): 1443-5756 c 2003 Victoria University. All rights reserved.
060-03
s(A) ≤
√
2||A||.
B OGDAN D. S UCEAVA˘
2
If A ∈ Mn (R), then: 12 1 2 (1.3) s(A) ≤ 2 1 − (trA) − 4E2 (A) , n with equality if and only if n − 2 of the characteristic roots of A are equal to the arithmetic mean of the remaining two. Consider now an isometrically immersed submanifold M n of dimension n ≥ 2 in a Rie¯ n+s , g¯). Then the Gauss and Weingarten formulae are given by mannian manifold (M ¯ X Y = ∇X Y + h(X, Y ), ∇ ¯ X ξ = −Aξ X + DX ξ, ∇ for every X, Y ∈ Γ(T M ) and ξ ∈ Γ(νM ). Take a vector η ∈ νp M and consider the linear mapping Aη : Tp M → Tp M. Let us consider the eigenvalues λ1η , . . . , λnη of Aη . We put (1.4)
Lη (p) = sup (λiη ) − inf (λiη ). i=1,...,n
i=1,...,n
Lη is the spread of the shape operator in the direction η. We define the spread of the shape operator at the point p by L(p) = sup Lη (p).
(1.5)
η∈νp M
¯. Suppose M is a compact submanifold of M 2 Let us remark that when M is a surface we have L2ν (p) = (λ1ν (p) − λ2ν (p))2 = 4(|H(p)|2 − K(p)), where ν is the normal vector at p, H is the mean curvature, and K is the Gaussian curvature. In [1] it is proved that for a surface M 2 in E2+s the geometric quantity (|H|2 − K)dV is a conformal invariant. As a corollary, one obtains for an orientable surface in E2+s that L2ν dV is a conformal invariant. Let ξn+1 , . . . , ξn+s be an orthonormal frame in the normal fibre bundle νM. Let us recall the definition of the extrinsic scalar curvature from [2]: s X X 2 ext = λin+r λjn+r . n(n − 1) r=1 i
