Lemma 3.2 (00VZ) · ScalingStacks

00VZ

Lemma 3.2. There is a uniform constant $C$ so that for any $0<t\leq 1$ , for any $y\in Y\backslash f(S)$ and for any $u\in C^{\infty}(X_{y})$ we have

(3.3)

\left(\int_{X_{y}}|u|^{\frac{2(n-m)}{n-m-1}}\omega_{y}^{n-m}\right)^{\frac{n-m-1}{n-m}}\leq C\int_{X_{y}}(|\nabla u|^{2}_{\omega_{y}}+|u|^{2})\omega_{y}^{n-m}.

00W0

Proof. For any $y\in Y\backslash f(S)$ the fiber $X_{y}$ is a smooth $(n-m)$ -dimensional complex submanifold of $X$ . Since $X$ is Kähler, it follows that $X_{y}$ is a minimal submanifold, and so it has vanishing mean curvature vector. We then use the Nash embedding theorem to isometrically embed $(X,\omega_{X})$ into Euclidean space, and so we have an isometric embedding $X\to\mathbb{R}^{N}$ . The length of the mean curvature vector of the composite isometric embedding $X_{y}\to X\to\mathbb{R}^{N}$ is then uniformly bounded independent of $y$ , since it depends only on the second fundamental form of $X\to\mathbb{R}^{N}$ . Then (3.3) follows from the uniform Sobolev inequality of [A, MS]. Notice that they prove an $L^{1}$ Sobolev inequality, but this implies the stated $L^{2}$ Sobolev inequality thanks to the Hölder inequality. ∎