Minimal surfaces with the area growth of two planes: the case of infinite symmetry.

*(English)*Zbl 1115.53008Consider a properly immersed minimal surface \(M\) in \(\mathbb E^3\) with area \(A(r)\) in balls \(B(r)\) of radius \(r\) centered at the origin. By the monotonicity formula, the function \(\overline A(r) =\frac{ A(r)}{r^2}\) is monotonically increasing. We say that \(M\) has area growth constant \(A(M) \in (0,\infty],\) if \(A(M) = \lim_{r\rightarrow \infty}\overline A(r).\) We say that M has quadratic area growth, if \(A(M) < \infty.\) Basic results in geometric measure theory imply that for any \(M\) with quadratic area growth and for any sequence of positive numbers \(t_i\rightarrow 0,\) the sequence of homothetic scalings \(M(i) = t_iM\) of \(M\) contains a subsequence that converges on compact subsets of \(\mathbb E^3\) to a limit minimal cone \(C\) in \(\mathbb E^3\) over a geodesic integral varifold in the unit sphere \(\mathbb S^2,\) which consists of a balanced finite configuration of geodesic arcs with positive integer multiplicities. \(C\) is called a limit tangent cone at infinity to \(M.\)

In 1834, H. F. Scherk [J. Reine Angew. Math., 13, 185–208 (1835; ERAM 013.0481cj)] discovered a singly-periodic embedded minimal surface \(S_{\frac {\pi}2}\) in \(\mathbb E^3\) with quadratic area growth constant \(2\pi.\) More generally, H. F. Scherk [loc. cit.] (also see H. Karcher [Manuscr. Math., 62,83–114, (1988; Zbl 0658.53006)]) defined a one-parameter deformation \(S_{\theta},\;\theta\in (0, \frac{\pi}2 ],\) of his original surface \(S_{\frac {\pi}2},\) which are also called Scherk surfaces and which may be viewed as the desingularization of two vertical planes with an angle \(\theta\) between them. The limit tangent cone at infinity to \(S_{\theta}\) consists of the union of these planes. In Volume 8 of [S. T. Yau, Surveys in Differential Geometry. Surveys in Differential Geometry 8. Somerville, MA: International Press. (2003; Zbl 1034.53003)] Meeks presented the following two conjectures related to minimal surfaces with quadratic area growth.

Conjecture 1 (Unique Limit Tangent Cone Conjecture). A properly immersed minimal surface in \(\mathbb E^3\) of quadratic area growth has a unique limit tangent cone at infinity.

Conjecture 2 (Scherk Uniqueness Conjecture). A connected properly immersed minimal surface \(M\) in \(\mathbb E^3\) with quadratic area growth constant \(A(M) < 3\pi\) must be a plane, a catenoid or a Scherk singly-periodic minimal surface \(S_{\theta},\) for some \(\theta\in (0, \frac{\pi}2].\)

The main goal of this paper is to prove Conjecture 2 under the additional hypothesis of infinite symmetry.

In 1834, H. F. Scherk [J. Reine Angew. Math., 13, 185–208 (1835; ERAM 013.0481cj)] discovered a singly-periodic embedded minimal surface \(S_{\frac {\pi}2}\) in \(\mathbb E^3\) with quadratic area growth constant \(2\pi.\) More generally, H. F. Scherk [loc. cit.] (also see H. Karcher [Manuscr. Math., 62,83–114, (1988; Zbl 0658.53006)]) defined a one-parameter deformation \(S_{\theta},\;\theta\in (0, \frac{\pi}2 ],\) of his original surface \(S_{\frac {\pi}2},\) which are also called Scherk surfaces and which may be viewed as the desingularization of two vertical planes with an angle \(\theta\) between them. The limit tangent cone at infinity to \(S_{\theta}\) consists of the union of these planes. In Volume 8 of [S. T. Yau, Surveys in Differential Geometry. Surveys in Differential Geometry 8. Somerville, MA: International Press. (2003; Zbl 1034.53003)] Meeks presented the following two conjectures related to minimal surfaces with quadratic area growth.

Conjecture 1 (Unique Limit Tangent Cone Conjecture). A properly immersed minimal surface in \(\mathbb E^3\) of quadratic area growth has a unique limit tangent cone at infinity.

Conjecture 2 (Scherk Uniqueness Conjecture). A connected properly immersed minimal surface \(M\) in \(\mathbb E^3\) with quadratic area growth constant \(A(M) < 3\pi\) must be a plane, a catenoid or a Scherk singly-periodic minimal surface \(S_{\theta},\) for some \(\theta\in (0, \frac{\pi}2].\)

The main goal of this paper is to prove Conjecture 2 under the additional hypothesis of infinite symmetry.

Reviewer: Doan The Hieu (Hue)

##### MSC:

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

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

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