Sharp upper bounds for Steklov eigenvalues of a hypersurface of revolution with two boundary components in Euclidean space (Q6618837)

The author provides some results on sharp upper bounds for the Steklov eigenvalues of \((M=[0,L]\times \mathbb S^{n-1},g_1)\), the hypersurface of revolution in Euclidean space with two boundary components each isometric to \(\mathbb S^{n-1}\) and meridian length \(L\). First, he states that there is a metric of revolution \(g_2\) on \(M\) defined as \(g_2(r,p)=dr^2+h^2(r)g_0(p)\) for \((r,p)\in [0,L]\times\mathbb S^{n-1}\) such that \(h\) is a positive smooth function on \([0,L]\) and satisfying \(h(0)=h(L)=1\), \(|h'(r)|\le 1\), and \(g_0\) is the metric defined on \(\mathbb S^{n-1}\) such that for each \(k\ge 1\), the \(kth\)-Steklov eigenvalue of \((M,g_1)\) is less than that of \((M,g_2)\) (Theorem 2). Next, the author states that an upper bound of \(\sigma_1(M)\) is given by \(B_n(L)\), an explicit function furnished in terms of \(n\) and \(L\) (Theorem 3). We say that \(L_1\) is the critical length if it is the unique positive solution of \(\displaystyle \left(1+\frac{L}{2}\right)^{2n-2}-(n-1)\left(1+\frac{L}{2}\right)^n-(n-1)^2\left(1+\frac{L}{2}\right)^{n-2}+n-1=0.\) Then, he states that for \(L\neq L_1\), there is a constant \(C(n,L)\) such that for all metric of revolutions \(g\) on \(M\), we have \(B_n-\sigma_1(M)\ge C(n,L)\). Furthermore, there exists a constant \(C(n)>0\) such that for all \(\delta\in \left(0,\displaystyle \frac{B_n-(n-2)}{2}\right)\), we have that \(|B_n-\sigma_1(M)|<\delta\) implies \(|L_1-L|<C(n)\delta\) (Theorem 6). Also, the author furnishes quantitative and qualitative results on critical lengths (Theorem 9).