The Michigan Mathematical Journal

A Half-Space Theorem for Ideal Scherk Graphs in $M\times\mathbb{R}$

Ana Menezes

Article information

Source
Michigan Math. J., Volume 63, Issue 4 (2014), 675-685.

Dates
First available in Project Euclid: 5 December 2014

https://projecteuclid.org/euclid.mmj/1417799220

Digital Object Identifier
doi:10.1307/mmj/1417799220

Mathematical Reviews number (MathSciNet)
MR3286665

Zentralblatt MATH identifier
1309.53005

Citation

Menezes, Ana. A Half-Space Theorem for Ideal Scherk Graphs in $M\times\mathbb{R}$. Michigan Math. J. 63 (2014), no. 4, 675--685. doi:10.1307/mmj/1417799220. https://projecteuclid.org/euclid.mmj/1417799220

References

• [1] U. Abresch and H. Rosenberg, Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28.
• [2] P. Collin and H. Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs, Ann. of Math. (2) 172 (2010), 1879–1906.
• [3] M. Dajczer and J. Ripoll, An extension of a theorem of Serrin to graphs in warped products, J. Geom. Anal. 15 (2005), no. 2, 193–205.
• [4] B. Daniel and L. Hauswirth, Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 445–470.
• [5] B. Daniel, W. H. Meeks, and H. Rosenberg, Half-space theorems for minimal surfaces in Nil3 and Sol3, J. Differential Geom. 88 (2011), 41–59.
• [6] P. Eberlein, Geodesic flows on negatively curved manifolds II, Trans. Amer. Math. Soc. 178 (1973), 57–82.
• [7] P. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Math., University of Chicago Press, Chicago, IL, 1996.
• [8] P. Eberlein and B. O’Neil, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109.
• [9] J. A. Gálvez and H. Rosenberg, Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces, Amer. J. Math. 132 (2010), no. 5, 1249–1273.
• [10] L. Hauswirth, H. Rosenberg, and J. Spruck, On complete mean curvature $\frac{1}{2}$ surfaces in $\mathbb{H}^{2}\times\mathbb{R}$, Comm. Anal. Geom. 16 (2008), no. 5, 989–1005.
• [11] D. Hoffman and W. H. Meeks III, The strong halfspace theorem for minimal surfaces, Invent. Math. 101 (1990), no. 2, 373–377.
• [12] L. Mazet, A general halfspace theorem for constant mean curvature surfaces, Amer. J. Math. 35 (2013), 801–834.
• [13] B. Nelli and H. Rosenberg, Minimal surfaces in $\mathbb{H}^{2}\times\mathbb{R}$, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 263–292.
• [14] B. Nelli and H. Rosenberg, Errata: “Minimal surfaces in $\mathbb{H}^{2}\times\mathbb{R}$” [Bull. Braz. Math. Soc. (N.S.), 33(2002), no 2, 263–292], Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 4, 661–664.
• [15] B. Nelli and R. Sa Earp, A halfspace theorem for mean curvature $H=\frac{1}{2}$ surfaces in $\mathbb{H}^{2}\times\mathbb{R}$, J. Math. Anal. Appl. 365 (2010), no. 1, 167–170.
• [16] H. Rosenberg, F. Schulze, and J. Spruck, The half-space property and entire positive minimal graphs in $M\times\mathbb{R}$, J. Differential Geom. 95 (2013), 321–336.
• [17] R. Schoen, Estimates for stable minimal surfaces in three dimensional manifolds, Ann. of Math. Stud. 103 (1983), 127–146.