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Minimal surfaces with the area growth of two planes: the case of infinite symmetry. (English) Zbl 1115.53008
Consider 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.

##### 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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##### References:
 [1] D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373 – 377. · Zbl 0722.53054 [2] H. Karcher, Embedded minimal surfaces derived from Scherk’s examples, Manuscripta Math. 62 (1988), no. 1, 83 – 114. · Zbl 0658.53006 [3] H. Karcher. Construction of minimal surfaces. Surveys in Geometry, pages 1-96, 1989. University of Tokyo, 1989, and Lecture Notes No. 12, SFB256, Bonn, 1989. [4] Hippolyte Lazard-Holly and William H. Meeks III, Classification of doubly-periodic minimal surfaces of genus zero, Invent. Math. 143 (2001), no. 1, 1 – 27. · Zbl 1031.53018 [5] Hai-Ping Luo. Desingularizing the intersection between a catenoid and a plane. Ph.D. thesis, University of Massachusetts, Amherst, 1997. [6] William H. Meeks III, Geometric results in classical minimal surface theory, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 269 – 306. · Zbl 1063.53008 [7] William H. Meeks III, Global problems in classical minimal surface theory, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 453 – 469. · Zbl 1100.53012 [8] William H. Meeks III and Joaquín Pérez, Conformal properties in classical minimal surface theory, Surveys in differential geometry. Vol. IX, Surv. Differ. Geom., vol. 9, Int. Press, Somerville, MA, 2004, pp. 275 – 335. · Zbl 1086.53007 [9] William H. Meeks III, Joaquín Pérez, and Antonio Ros, Uniqueness of the Riemann minimal examples, Invent. Math. 133 (1998), no. 1, 107 – 132. · Zbl 0916.53004 [10] William H. Meeks III and Harold Rosenberg, The global theory of doubly periodic minimal surfaces, Invent. Math. 97 (1989), no. 2, 351 – 379. · Zbl 0676.53068 [11] William H. Meeks III and Harold Rosenberg, The maximum principle at infinity for minimal surfaces in flat three manifolds, Comment. Math. Helv. 65 (1990), no. 2, 255 – 270. · Zbl 0713.53008 [12] William H. Meeks III and Harold Rosenberg, The geometry of periodic minimal surfaces, Comment. Math. Helv. 68 (1993), no. 4, 538 – 578. · Zbl 0807.53049 [13] Joaquín Pérez, M. Magdalena Rodríguez, and Martin Traizet, The classification of doubly periodic minimal tori with parallel ends, J. Differential Geom. 69 (2005), no. 3, 523 – 577. · Zbl 1094.53007 [14] J. Pérez and M. Traizet. The classification of singly periodic minimal surfaces with genus zero and Scherk type ends. Transactions of the A.M.S. To appear. · Zbl 1110.53008 [15] H. F. Scherk. Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen. J. R. Angew. Math., 13:185-208, 1835. · ERAM 013.0481cj [16] Richard M. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), no. 4, 791 – 809 (1984). · Zbl 0575.53037 [17] Martin Traizet, Weierstrass representation of some simply-periodic minimal surfaces, Ann. Global Anal. Geom. 20 (2001), no. 1, 77 – 101. · Zbl 1033.53008 [18] Martin Traizet, An embedded minimal surface with no symmetries, J. Differential Geom. 60 (2002), no. 1, 103 – 153. · Zbl 1054.53014 [19] M. Weber and M. Wolf, Minimal surfaces of least total curvature and moduli spaces of plane polygonal arcs, Geom. Funct. Anal. 8 (1998), no. 6, 1129 – 1170. · Zbl 0954.53007 [20] Matthias Weber and Michael Wolf, Teichmüller theory and handle addition for minimal surfaces, Ann. of Math. (2) 156 (2002), no. 3, 713 – 795. · Zbl 1028.53009 [21] Michael Wolf, Flat structures, Teichmüller theory and handle addition for minimal surfaces, Global theory of minimal surfaces, Clay Math. Proc., vol. 2, Amer. Math. Soc., Providence, RI, 2005, pp. 211 – 241. · Zbl 1109.53010
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