Inequalities of Willmore type for submanifolds
Let \(M^ n\) be a compact smooth manifold, \(f: M^ n\to {\mathbb{R}}^ m\) an immersion, N(f) the unit normal bundle of f. To every \(\xi \in N_ p(f)\) there corresponds a shape operator \(A_{\xi}\) with eigenvalues \(k_ 1(\xi),...,k_ n(\xi)\). Put \(\sigma (A_{\xi})^ 2=(1/n^ 2)| \sum_{i<j}(k_ i-k_ j)^ 2.\) Then the total conformal curvature C(f) of the immersion f is defined as \[ C(f)=\int_{N(f)}\sigma (A_{\xi})^ n d\xi /Vol(S^ m). \] Here \(d\xi\) is the natural volume element on N(f). For \(n=2\), we have \[ C(f)=(1/2\pi)\int_{M}(| H|^ 2\quad -\quad K)dA \] and \(C(f)\geq \beta_ 1\) is the first \({\mathbb{Z}}_ 2\)-Betti number of \(m^ 2\). In the present paper, the author proves the following generalization of this inequality: Let \(M^ n\) be compact, F a field, \(\beta_ 1,...,\beta_ n\) the Betti numbers of \(M^ n\) with respect to F, \(f: M^ n\to {\mathbb{R}}^ m\) an immersion. Then \[ C(f)\geq \sum^{n-1}_{k=1}a_ k\beta_ k,\text{ where } a_ k=(k/n-k)^{n/2-k}. \] The Willmore conjecture asserts for all immersed tori \(T^ 2\) in \({\mathbb{R}}^ 3\) the inequality: \(\int_{T^ 2}H^ 2 dA\geq 2\pi^ 2\). The present author considers the following problems: (a) For a given compact smooth manifold \(M^ n\), determine \(C(M^ n):=\inf \{C(f)| \quad f: M^ n\to {\mathbb{R}}^ m\quad an\quad immersion\}\). (b) For \(n\geq 2\), determine (or estimate) \(C(n):=\inf \{C(M^ n)| \quad M^ n\quad not\quad \hom eomorphic\quad to\quad S^ n\}\). For \(n=2\), \textit{P. Li} and \textit{S.-T. Yau} [Invent. Math. 69, 269-291 (1982; Zbl 0503.53042)] give the complete answer \(C(2)=C({\mathbb{R}}P^ 2)=2\). The author shows C(n)\(\geq 2(n-1)^{1-n/2}\) in this paper.
scientific article; zbMATH DE number 3971672