The slab theorem for minimal surfaces in $$\mathbb {E}(-1,\tau )$$
Tóm tắt
Unlike
$$\mathbb {R}^{3}$$
, the homogeneous spaces
$$\mathbb {E}(-1,\tau )$$
have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantee that a minimal surface in
$$\mathbb {E}(-1,\tau )$$
is such a graph. More specifically, we introduce the definition of a generalized slab in
$$\mathbb {E}(-1,\tau )$$
and prove that a properly immersed minimal surface of finite topology inside such a slab region has multi-graph ends. Moreover, when the surface is embedded, the ends are graphs. When the surface is embedded and simply connected, it is an entire graph.
Tài liệu tham khảo
Bernstein, J., Brenner, C.: Conformal structure of minimal surfaces with Finite topology. Com- ment. Math. Helv. 86(2), 353–381 (2011)
Colding, T.H., Minicozzi II, W.P.: The space of embedded minimal surfaces of fixed genus in a 3-manifold. I. Estimates off the axis for disks. Ann. Math. (2) 160(1), 27–68 (2004)
Colding, T.H., Minicozzi II, W.P.: The Calabi–Yau conjectures for embedded surfaces. Ann. Math. (2) 167(1), 211243 (2008)
Collin, P.: Topologie et courbure des surfaces minimales proprement plongées de \({\mathbb{R}^{3}}\). Ann. Math. 145, 1–31 (1997)
Collin, P., Hauswirth, L., Rosenberg, H.: Minimal surfaces in finite volume hyperbolic 3-manifolds and \(M\times {\mathbb{S}}^{1}\), \(M\) a finite area hyperbolic surface. arXiv:1304.1773
Collin, P., Hauswirth, L., Rosenberg, H.: Properly immersed minimal surfaces in a slab of \(\mathbb{H}\times \mathbb{R}\), \(\mathbb{H}\) the hyperbolic plane. Arch. Math. (Basel) 104(5), 471–484 (2015)
Collin, P., Rosenberg, H.: Construction of harmonic diffeomorphisms and minimal graphs. Ann. Math. 172(3), 1879–1906 (2010)
Coskunuzer, B.: Non-properly embedded minimal planes in hiperbolic 3-space. Commun. Contemp. Math. 13, 727–739 (2011)
Folha, A., Peñafiel, C.: Minimal graphs in \(\widetilde{{\rm PSL}_{2}}({\mathbb{R}}, \tau )\) (preprint)
Hoffman, D., Meeks III, W.H.: The strong halfspace theorem for minimal surfaces. Invent. Math. 101, 373–377 (1990)
Lima, V.: The Topology, Conformal Structure and Geometry of Minimal and CMC Surfaces in Riemannian Products and Homogeneous 3-Manifolds. Phd Thesis, Instituto Nacional de Matemática Pura e Aplicada—IMPA (2015)
Mazet, L.: The half space property for cmc 1/2 graphs in \(\mathbb{E}(-1,\tau )\). Calc. Var. Partial Differ. Equ. 52(3–4), 661–680 (2015)
Meeks III, W.H., Rosenberg, H.: The uniqueness of the helicoid. Ann. Math. 161, 727–758 (2005)
Melo, S.: Minimal graphs in \(\widetilde{{\rm PSL}_{2}(\mathbb{R})}\) over unbounded domains. Bull. Braz. Math. Soc. New Ser. 45(1), 91–116 (2014)
Nelli, B., Rosenberg, H.: Minimal surfaces in \(\mathbb{H}^{2}\times \mathbb{R}\). Bull. Braz. Math. Soc. (N.S.) 33(2), 263–292 (2002)
Nelli, B., Rosenberg, H.: Errata: minimal surfaces in \({\mathbb{H}}^{2}\times {\mathbb{R}}\), [Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 2, 263292; MR1940353]. Bull. Braz. Math. Soc. (N.S.) 38(4) (2007), 661664 (53A10)
Peñafiel, C.: Invariant surfaces in \(\widetilde{{\rm PSL}_{2}}(\mathbb{R},\tau )\) and applications. Bull. Braz. Math. Soc. (Impresso) 43, 545–578 (2011)
Rodríguez, M., Tinaglia, G.: Nonproper complete minimal surfaces embedded in \({\mathbb{H}}^{2}\times {\mathbb{R}}\). Int. Math. Res. Not. IMRN 2015(12), 4322–4334 (2015)
Younes, R.: Minimal surfaces in \(\widetilde{{\rm PSL}_{2}({\mathbb{R}})}\). Ill. J. Math. 54(2), 671712 (2010)