##
**Minimal annuli with and without slits.**
*(English)*
Zbl 1087.53008

From the introduction: We bound the oscillation of the unit normal of minimal annuli with and without slits. Our estimates are independent of the ratio of the inner and outer radii. Hence, we recover standard removable singularity results as the inner radius goes to zero. The estimate for annuli with slits is important in proving a removable singularities theorem for minimal limit laminations.

Proposition 1.3 shows that if a minimal annulus \(\Sigma\) in \(\mathbb R^3\) has \(\int_{\partial\Sigma} |A|<\pi/8\) and \(\int_\Sigma K\geq-\pi\), then \(\Sigma\) is a graph. Proposition 1.12 extends this to surfaces with quasi-conformal Gauss map and shows that if \(\int_{\partial\Sigma} |A|\) is actually small, then \(\Sigma\) is Lipschitz close to a plane.

In Section 3 we extend this to what we call “minimal annuli with slits”. These are multi-valued minimal graphs over an annulus in the plane.

In Theorem 3.36, we obtain a similar bound in general for minimal annuli with slits satisfying certain boundary conditions. This bound implies that there is a fixed plane which these are Lipschitz close to on every scale.

The results given here should be compared with Rado’s theorem [see, for instance, T. H. Colding and W. P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in Mathematics. 4. New York, NY: Courant Institute of Mathematical Sciences (1999; Zbl 0987.49025)] which states that a minimal surface in \(\mathbb R^3\) whose bounday is a circle which is a graph over the boundary of a convex set in a plane is itself a graph (and in fact a disk).

Proposition 1.3 shows that if a minimal annulus \(\Sigma\) in \(\mathbb R^3\) has \(\int_{\partial\Sigma} |A|<\pi/8\) and \(\int_\Sigma K\geq-\pi\), then \(\Sigma\) is a graph. Proposition 1.12 extends this to surfaces with quasi-conformal Gauss map and shows that if \(\int_{\partial\Sigma} |A|\) is actually small, then \(\Sigma\) is Lipschitz close to a plane.

In Section 3 we extend this to what we call “minimal annuli with slits”. These are multi-valued minimal graphs over an annulus in the plane.

In Theorem 3.36, we obtain a similar bound in general for minimal annuli with slits satisfying certain boundary conditions. This bound implies that there is a fixed plane which these are Lipschitz close to on every scale.

The results given here should be compared with Rado’s theorem [see, for instance, T. H. Colding and W. P. Minicozzi II, Minimal surfaces, Courant Lecture Notes in Mathematics. 4. New York, NY: Courant Institute of Mathematical Sciences (1999; Zbl 0987.49025)] which states that a minimal surface in \(\mathbb R^3\) whose bounday is a circle which is a graph over the boundary of a convex set in a plane is itself a graph (and in fact a disk).

### MSC:

53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |

49Q05 | Minimal surfaces and optimization |