6. Estimates for injectivity radius [05E6]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

6. Estimates for injectivity radius

In [21], Harvey and Lawson introduced the notion of calibrated submanifold. If $(M,g)$ is a Riemannian manifold, and $\Theta$ is a closed $n$ -form such that $\Theta|_{\xi}\leq dv_{\xi}$ for any oriented $n$ -plane $\xi$ in the tangent bundle of $M$ , then $\Theta$ is called a calibration on $M$ , where $dv_{\xi}$ denotes the volume form on $\xi$ . An oriented $n$ -submanifold $L$ of $M$ is called calibrated by the calibration $\Theta$ , if $\Theta|_{L}$ equals to the volume form of $g|_{L}$ on $L$ . Mclean studied the deformation theory of calibrated submanifolds in [28].

Holomorphic submanifolds in Kähler manifolds, and special lagrangian submanifolds in Calabi-Yau manifolds are examples of calibrated submanifolds (c.f. [21]). If $(M,\omega,J,g)$ is a Kähler $n$ -manifold, then $\frac{1}{m!}\omega^{m}$ , $m\leq n$ , are calibrations on $M$ , and holomorphic $m$ -submanifolds are calibrated by $\frac{1}{m!}\omega^{m}$ . If $(M,\omega,J,g,\Omega)$ is a Ricci-flat Calabi-Yau $n$ -manifold, then, for any $\theta\in\mathbb{R}$ , ${\rm Re}e^{\sqrt{-1}\theta}\Omega$ is a calibration on $M$ , and a special lagrangian submanifold $L$ of phase $\theta$ is calibrated by ${\rm Re}e^{\sqrt{-1}\theta}\Omega$ .

In [16], a volume comparison theorem for calibrated submanifolds was obtained.

Theorem 6.1 (Theorem 2.0.1. in [16]).

Let $(M,g)$ be a closed Riemannian manifold, $\Theta$ be a calibration n-form, and $p\in M$ . Assume that the sectional curvature $K_{g}$ satisfies

\sup_{B_{g}(p,2\pi)}K_{g}\leq 1,

and there is a submanifold $L$ calibrated by $\Theta$ such that $p\in L$ . Then

Vol_{g}(B_{g}(p,r)\cap L)\geq Vol_{h_{1}}(B_{h_{1}}(r)),

for any $r\leq\min\{i_{g}(p),\pi\}$ , where $h_{1}$ denotes the standard metric on $S^{n}$ with constant curvature 1, and $B_{h_{1}}(r)$ denotes a metric $r$ -ball in $S^{n}$ .

By this theorem, we obtain the following estimate for injectivity radius:

Corollary 6.2.

Let $(M,g)$ be a closed Riemannian manifold, $\Theta$ be a calibration n-form, and $p\in M$ . Assume that the sectional curvature $K_{g}$ satisfies

\sup_{B_{g}(p,2\pi)}K_{g}\leq 1,

and there is a submanifold $L$ calibrated by $\Theta$ such that $\dim_{\mathbb{R}}L=n$ , $p\in L$ , and

\int_{L}\Theta<\frac{\pi}{2n}\varpi_{n-1},

where $\varpi_{n-1}$ is the volume of $S^{n-1}$ with the standard metric of constant curvature 1. Then the injectivity radius $i_{g}(p)$ of $(M,g)$ at $p$ satisfies that

i_{g}(p)^{n}\leq\frac{n\pi^{n-1}}{2^{n-1}\varpi_{n-1}}\int_{L}\Theta.

Proof.

By Theorem 6.1, we have

Vol_{h_{1}}(B_{h_{1}}(r))\leq Vol_{g}(B_{g}(p,r)\cap L)\leq Vol_{g}(L)=\int_{L}\Theta,

for any $r\leq\min\{i_{g}(p),\frac{\pi}{2}\}$ , where $h_{1}$ denotes the standard metric on $S^{n}$ with constant curvature 1, and $B_{h_{1}}(r)$ denotes a metric $r$ -ball in $S^{n}$ . Since $h_{1}=dr^{2}+\sin^{2}rh_{S^{n-1}}$ where $h_{S^{n-1}}$ is the standard metric on $S^{n-1}$ with constant curvature 1, we obtain $\sin r\geq\frac{2}{\pi}r$ , and

\frac{2^{n-1}}{n\pi^{n-1}}r^{n}\varpi_{n-1}\leq\int_{0}^{r}\sin^{n-1}rdr\varpi_{n-1}=Vol_{h_{1}}(B_{h_{1}}(r))\leq\int_{L}\Theta.

If $i_{g}(p)\geq\frac{\pi}{2}$ , by letting $r=\frac{\pi}{2}$ , we obtain

\frac{\pi}{2n}\varpi_{n-1}\leq\int_{L}\Theta<\frac{\pi}{2n}\varpi_{n-1},

which is a contradiction. Thus $i_{g}(p)<\frac{\pi}{2}$ . By letting $r=i_{g}(p)$ , we obtain

i_{g}(p)^{n}\leq\frac{n\pi^{n-1}}{2^{n-1}\varpi_{n-1}}\int_{L}\Theta.

∎

We obtain Theorem 1.5 by applying the above corollary to special lagrangian submanifolds in Ricci-flat Calabi-Yau manifolds. Another obvious application of Corollary 6.2 is to estimate injectivity radiuses by volumes of holomorphic submanifolds, which has independent interests.

Corollary 6.3.

Let $(M,\omega,J,g)$ be a closed Kähler n-manifold, and $p\in M$ . Assume that the sectional curvature $K_{g}$ satisfies

\sup_{M}K_{g}\leq 1,

and there is a smooth holomorphic m-submanifold $N$ such that $p\in N$ , and

\int_{N}\omega^{m}<\frac{(m-1)!\pi}{2}\varpi_{m-1}.

Then the injectivity radius $i_{g}(p)$ of $(M,g)$ at $p$ satisfies that

i_{g}(p)^{m}\leq\frac{\pi^{m-1}}{(m-1)!2^{m-1}\varpi_{m-1}}\int_{N}\omega^{m}.

By combining this corollary and the result in [6], there are $F$ -structures of positive rank on the regions of Kähler manifolds with bounded curvature and fibred by holomorphic submanifolds with small volumes.