caa a22 4500 605496102 CHVBK 20210128100534.0 cr unu---uuuuu 210128e20151201xx s 000 0 eng 10.1007/s10231-014-0446-y doi (NATIONALLICENCE)springer-10.1007/s10231-014-0446-y Constructions of $$H_r$$ H r -hypersurfaces, barriers and Alexandrov theorem in $$\mathrm{I\!H}^n\times \mathrm{I\!R}$$ I H n × I R [Elektronische Daten] [Maria Elbert, Ricardo Sa Earp] In this paper, we are concerned with hypersurfaces in $$\mathrm{I\!H}\times \mathrm{I\!R}$$ I H × I R with constant $$r$$ r -mean curvature, to be called $$H_r$$ H r -hypersurfaces. We construct examples of complete $$H_r$$ H r -hypersurfaces, which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $$H_r$$ H r -graph for each value $$0<H_r\le \frac{n-r}{n}$$ 0 < H r ≤ n - r n . Also, for each value $$H_r>\frac{n-r}{n}$$ H r > n - r n , there is a unique embedded compact strictly convex rotational $$H_r$$ H r -hypersurface. By using them as barriers, we obtain some interesting geometric results, including height estimates and an Alexandrov-type Theorem. Namely, we prove that an embedded compact $$H_r$$ H r -hypersurface in $$\mathrm{I\!H}^n\times \mathrm{I\!R}$$ I H n × I R is rotational ( $$H_r>0$$ H r > 0 ). Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 $$r$$ r -Mean curvature nationallicence Alexandrov Theorem nationallicence $$H_r$$ H r -Hypersurfaces nationallicence Barriers nationallicence Entire vertical graphs nationallicence Complete horizontal graphs nationallicence Elbert Maria Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil aut Sa Earp Ricardo Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/6(2015-12-01), 1809-1834 0373-3114 194:6<1809 2015 194 10231 https://doi.org/10.1007/s10231-014-0446-y text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0446-y text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Elbert Maria Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil aut NATIONALLICENCE 700 1- Sa Earp Ricardo Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/6(2015-12-01), 1809-1834 0373-3114 194:6<1809 2015 194 10231