Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space

Fernández, Isabel; Mira, Pablo

doi:10.1090/s0002-9947-09-04645-5

Cited by 38 publications

(65 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• in [DH09], the author and L. Hauswirth proved that every complete nowhere vertical minimal surface in Nil 3 is necessarily an entire graph; moreover, this implies using [FM09] that every complete nowhere vertical minimal surface in Nil 3 comes from a complete CMC surface in L 3 .…”

Section: Final Remarkmentioning

confidence: 97%

See 1 more Smart Citation

The Gauss Map of Minimal Surfaces in the Heisenberg Group

Daniel¹

2010

International Mathematics Research Notices

View full text Add to dashboard Cite

Abstract. We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H 2 . Conversely, any nowhere antiholomorphic harmonic map into H 2 is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of complete nowhere vertical minimal surfaces.

show abstract

Section: Final Remarkmentioning

confidence: 97%

“…• in [FM09], I. Fernández and P. Mira solved the Bernstein problem in Nil 3 , i.e., they classified all entire minimal graphs in Nil 3 (in terms of their Abresch-Rosenberg differential),…”

Section: Final Remarkmentioning

confidence: 99%

The Gauss Map of Minimal Surfaces in the Heisenberg Group

Daniel¹

2010

International Mathematics Research Notices

View full text Add to dashboard Cite

show abstract

“…(1) The topological classification of minimal surfaces in R 3 by Frohman and Meeks [57]. (2) The uniqueness of Scherk's singly periodic minimal surfaces by Meeks and Wolf [143].…”

Section: Theorem 16 (Meeks Pérez Ros [119]) For Every Non-negativmentioning

confidence: 99%

The classical theory of minimal surfaces

Meeks¹,

Pérez²

2011

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane, the helicoid, the catenoid and the one-parameter family {R t } t∈(0,1) of Riemann minimal examples are the only complete, properly embedded, minimal planar domains in R 3 ; the proof of this result depends primarily on work of Colding and Minicozzi, Collin, López and Ros, Meeks, Pérez and Ros, and Meeks and Rosenberg. Rather than culminating and ending the theory with this classification result, significant advances continue to be made as we enter a new golden age for classical minimal surface theory. Through our telling of the story of the classification of minimal planar domains, we hope to pass on to the general mathematical public a glimpse of the intrinsic beauty of classical minimal surface theory and our own perspective of what is happening at this historical moment in a very classical subject.

show abstract

“…Theorem ( [FM07b]). Let Q be a holomorphic quadratic differential on D or a non-identically zero holomorphic quadratic differential on C. Then there exists a 2-parameter family of generically non-congruent entire minimal graphs in Nil 3 whose Abresch-Rosenberg differential is Q.…”

Section: Introductionmentioning

confidence: 99%

Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group

Daniel

Hauswirth

2008

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. We construct a one-parameter family of properly embedded minimal annuli in the Heisenberg group Nil 3 endowed with a left-invariant Riemannian metric. These annuli are not rotationally invariant. This family gives a vertical half-space theorem and proves that each complete minimal graph in Nil 3 is entire. Also, the sister surface of an entire minimal graph in Nil 3 is an entire constant mean curvature annuli in H 2 × R.

show abstract

Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space

Cited by 38 publications

References 17 publications

The Gauss Map of Minimal Surfaces in the Heisenberg Group

The Gauss Map of Minimal Surfaces in the Heisenberg Group

The classical theory of minimal surfaces

Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group

Contact Info

Product

Resources

About