TY - JOUR

T1 - On the characterization of minimal surfaces with finite total curvature in H 2 × R and PSL ~ 2 (R)

AU - Hauswirth, Laurent

AU - Menezes de Jesus, Ana Maria

AU - Rodríguez, Magdalena

PY - 2019/4/1

Y1 - 2019/4/1

N2 - It is known that a complete immersed minimal surface with finite total curvature in H 2 × R is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg in Mat Contemp 31:65–80, 2006; Hauswirth et al. in Adv Math 274:199–240, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in H 2 × R. As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in H 2 × R. We also prove that if a properly immersed minimal surface in PSL ~ 2 (R, τ) has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.

AB - It is known that a complete immersed minimal surface with finite total curvature in H 2 × R is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg in Mat Contemp 31:65–80, 2006; Hauswirth et al. in Adv Math 274:199–240, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in H 2 × R. As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in H 2 × R. We also prove that if a properly immersed minimal surface in PSL ~ 2 (R, τ) has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.

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U2 - https://doi.org/10.1007/s00526-019-1505-4

DO - https://doi.org/10.1007/s00526-019-1505-4

M3 - Article

VL - 58

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 2

M1 - 80

ER -