Volume monotonicity and lower bound [04ES]

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Volume monotonicity and lower bound

Corollary 5.12.

(Volume lower bound) If $P$ is in the support of $L$ , then there is a uniform lower bound on the volume of $L$ inside Eulidean coordinate balls of radius less than $R$ :

\text{Vol}_{g}(L\cap B(P,r))\geq C^{-1}r^{n},\quad\forall r\leq R.

(56)

Proof.

Let $f(r)=\mathcal{H}^{n}(L\cap B(r))>0$ , then $f$ is increasing in $r$ , and for a.e. $0<r<R$ , by the coarea formula,

f^{\prime}(r)=\int_{\partial B(r)\cap L}\frac{1}{|\nabla r|}d\mathcal{H}^{n-1}\geq C^{-1}\mathcal{H}^{n-1}(\partial B(r)\cap L)\geq C^{-1}f(r)^{(n-1)/n}.

The last inequality is the isoperimetric inequality. Thus $\frac{d}{dr}f^{1/n}\geq C^{-1},$ whence we have the volume lower bound $f(r)^{1/n}\geq C^{-1}r$ . ∎

Remark 5.12.

Volume lower bounds like (56) are familiar in minimal surface theory, but usually require some integral bound on the mean curvature. Notably, here we need no such assumption; the quantitative almost calibrated condition only concerns the antiderivative $\theta$ of $\vec{H}=J\nabla\theta$ .