Lower bound estimate of a series of Willmore type functionals in terms of Betti numbers of submanifolds (Q2704137)

Let \(f:M^n\to E^{n+p}\) be an isometric immersion from an \(n\)-dimensional compact Riemannian manifold \(M^n\) into the Euclidean \((n+p)\)-space \(E^{n+p}\). Denote by \(NM\to M\) the normal bundle of \(f\) with the induced metric by \(f\). Let \(BM\) be the unit normal bundle over \(M\). For any point \((x,e)\in BM\) where \(x\in M,e\in T^{\bot}M\), the corresponding shape operator \(A_e:T_xM\to T_xM\) of \(M\) has eigenvalues \(h_1(x,e),\cdots ,h_n(x,e)\) called the principal curvatures of \(M\) with respect to \(e\). The author of this paper considers the quantities: NEWLINE\[NEWLINE\sigma^{(2r)}(A_e)=\frac 1{n^2}\sum_{i<j}(h_i(x,e)-h_j(x,e))^{2r}NEWLINE\]NEWLINE for \(r=1,2,\cdots ,[n/2]\), and defines the following functionals: NEWLINE\[NEWLINEC^{(2r)}(M,f)=\frac 1{C_{n+p-1}}\int_{BM}(\sigma_{(2r)}(A_e))^{\frac n{2r}}d\mu,NEWLINE\]NEWLINE where \(d\mu\) denotes the volume element of \(BM\) and \(C_{n+p-1}\) is the volume of the unit \((n+p-1)\)-sphere. These are conformal invariants in \(E^{n+p}\). The main result of this paper is to give the following estimate of lower bounds: NEWLINE\[NEWLINEC^{(2r)}(M,f)\geq \sum_{k=1}^{n-1}B^{(2r)}(n,k)\beta_k,NEWLINE\]NEWLINE where \(B^{(2r)}(n,k)=(\frac kn)^{\frac n{2r}-k}(\frac {n-k}n)^{\frac n{2r}-n+k}\) and \(\beta_k\) denotes the \(k\)-th Betti number of \(M\). The case of \(r=1\) was obtained by \textit{U. Pinkall} in [Math. Z. 193, 241-246 (1986; Zbl 0602.53039)].