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Dear all,

a question came to me when I read the paper "Complete three dimensional manifolds with positive Ricci curvature and scalar curvature/ R. Schoen and S.-T. Yau, 1982".

The question is as follows. suppose we have a sequence of complete non-compact smooth submanifolds $\Sigma_k\subset M$ and each $\Sigma_k$ is area-minimizing (in its isotopy class) with respect to any compactly supported deformation. Then by treating them as Radon measures, we can get a convergent subsequence of $\Sigma_k$, say $\Sigma_k\to\Sigma$. The convergence is of the sense of Radon measure, or is called "in the sense of varifold". As I know, it is merely stronger than Cheeger-Gromov convergence. However, in Schoen-Yau's article, they say that the convergence is smooth. (page 217, the last paragraph in the proof of Lemma 3.) I cannot see the reason for this convergence to be smooth.

Maybe they have used some other properties in that paper (I guess not), but it's difficult for me to provide all the details here.... Anyway, thank you all for any comments.

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The reason for this is Allard's regularity theorem.

Roughly speaking Allard's theorem says that if near a point of the support of a stationary varifold the varifold has unit density and area close to that of the ball of the appropriate dimension (for a 2-varifold it would be area of a disk) then the support of the varifold is smooth at that point.

More precisely, there is an $\epsilon>0$ and $r_0>0$ (depending on the ambient geometry and dimension of the varifold). So that if $\Sigma$ is an stationary $m$-varifold in $M$ , a Riemannian manifold, and a point $p\in spt \Sigma$ satisfies $$\mathcal{H}^m(B_r(p)\cap \Sigma) \leq \omega_m r^m(1+\epsilon)$$ for $r\leq r_0 $ then $spt \Sigma$ is smooth near $p$. Here $B_r(p)$ is the $r$-ball in $M$ and $\omega_m$ is the volume of the unit ball in $\mathbb{R}^m$.

Allard's proof is unfortunately quite technical -- a good reference is Leon Simon's (sadly) hard to find book "Lectures on Geometric Measure Theory". Luckily, for your purpose there is a simpler version with a very easy proof due to Brian White (see here for the paper).

Specifically, suppose you have instead of being a stationary varifold you know that $\Sigma$ is a smooth minimal surface. If $p$ is a point of $\Sigma$ so that $$\mathcal{H}^m(B_r(p)\cap \Sigma) \leq \omega_m r^m(1+\epsilon)$$ then one has $$|A|(p)\leq r^{-1}$$ here $|A|$ is the norm of the second fundamental form. In other words you obtain a quantitive bound on curvature.

How does this relate to your question? Well you have that $\Sigma_k$ converge to $\Sigma$ as Radon measures. Let $p\in \Sigma$ this convergence implies that $$\mathcal{H}^m(\Sigma_k\cap B_r(p))\to \mathcal{H}^m(\Sigma\cap B_r(p))$$ (it is worth noting that we are using that the surfaces are area minimizing to ensure the convergence is with multiplicity one). Now since $\Sigma$ is smooth near $p$ it is locally modelled on a flat plane. In other words, there is a scale $r$ so that $$\mathcal{H}^m(B_r(p)\cap \Sigma) \leq \omega_m r^m(1+\frac{1}{2}\epsilon)$$ hence by the convergence $$\mathcal{H}^m(B_r(p)\cap \Sigma_k) \leq \omega_m r^m(1+\epsilon)$$ and so since the $\Sigma_k$ are smooth (either a priori or by Allard's full theorem) and the point $p$ was not important we deduce that for some $\delta>0$ we have $$\sup_{B_\delta(p)\cap\Sigma_k} |A|\leq r^{-1}.$$ In other words there is a uniform curvature bound on the $\Sigma_k$. The result then follows from "Standard elliptic PDE" and the Arzela-Ascoli theorem.

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  • $\begingroup$ Thanks, Rbega, your answer is extremely clear and solves my problem! Thanks a lot! $\endgroup$ – Chih-Wei Chen Sep 22 '11 at 1:35
  • $\begingroup$ Hi Rbega, it seems this argument requires that the limit $\Sigma$ is smooth, in the context of the Schoen-Yau paper how to show this? $\endgroup$ – Caramba Oct 4 '12 at 16:27
  • $\begingroup$ @Caramba You use the regularity theory for area-minimizing hypersurfaces. The issue Chih-Wei was asking about was why the convergence was smooth. $\endgroup$ – Rbega Oct 4 '12 at 18:33

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