Complete minimal surfaces of finite topology in the doubled Schwarzschild 3-manifold
-
Jaigyoung Choe
Abstract
We construct complete embedded minimal surfaces of arbitrary genus in the doubled Schwarzschild 3-manifold. A classical desingularization method is used for the construction.
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: NRF-RS-2023-00246133
Award Identifier / Grant number: NRF-2022R1C1C2013384
Award Identifier / Grant number: NRF-2021R1A4A1032418
Funding source: Korea Institute for Advanced Study
Award Identifier / Grant number: MG086401
Funding statement: Jaigyoung Choe supported in part by NRF-RS-2023-00246133; Jaehoon Lee supported in part by a KIAS Individual Grant MG086401 at Korea Institute for Advanced Study; Eungbeom Yeon supported in part by National Research Foundation of Korea NRF-2022R1C1C2013384 and NRF-2021R1A4A1032418.
References
[1] F. J. Almgren, Jr. and L. Simon, Existence of embedded solutions of Plateau’s problem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 6 (1979), no. 3, 447–495. Search in Google Scholar
[2] Y. Bernard and T. Rivière, Ends of immersed minimal and Willmore surfaces in asymptotically flat spaces, Comm. Anal. Geom. 28 (2020), no. 1, 1–57. 10.4310/CAG.2020.v28.n1.a1Search in Google Scholar
[3] S. Brendle, Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 247–269. 10.1007/s10240-012-0047-5Search in Google Scholar
[4] A. Carlotto, Rigidity of stable minimal hypersurfaces in asymptotically flat spaces, Calc. Var. Partial Differential Equations 55 (2016), no. 3, 53–73. 10.1007/s00526-016-0989-4Search in Google Scholar
[5] A. Carlotto, O. Chodosh and M. Eichmair, Effective versions of the positive mass theorem, Invent. Math. 206 (2016), no. 3, 975–1016. 10.1007/s00222-016-0667-3Search in Google Scholar
[6] A. Carlotto and A. Mondino, A non-existence result for minimal catenoids in asymptotically flat spaces, J. Lond. Math. Soc. (2) 95 (2017), no. 2, 373–392. 10.1112/jlms.12012Search in Google Scholar
[7] O. Chodosh and D. Ketover, Asymptotically flat three-manifolds contain minimal planes, Adv. Math. 337 (2018), 171–192. 10.1016/j.aim.2018.08.010Search in Google Scholar
[8]
J. Choe and M. Soret,
New minimal surfaces in
[9] D. Hoffman, F. Martín and B. White, Nguyen’s tridents and the classification of semigraphical translators for mean curvature flow, J. reine angew. Math. 786 (2022), 79–105. 10.1515/crelle-2022-0005Search in Google Scholar
[10] D. Hoffman and W. H. Meeks, III, Embedded minimal surfaces of finite topology, Ann. of Math. (2) 131 (1990), no. 1, 1–34. 10.2307/1971506Search in Google Scholar
[11]
H. B. Lawson, Jr.,
Complete minimal surfaces in
[12] W. H. Meeks, III, L. Simon and S. T. Yau, Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659. 10.2307/2007026Search in Google Scholar
[13] W. H. Meeks, III and S. T. Yau, The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Douglas-Morrey and an analytic proof of Dehn’s lemma, Topology 21 (1982), no. 4, 409–442. 10.1016/0040-9383(82)90021-0Search in Google Scholar
[14] H. Rosenberg, R. Souam and E. Toubiana, General curvature estimates for stable 𝐻-surfaces in 3-manifolds and applications, J. Differential Geom. 84 (2010), no. 3, 623–648. 10.4310/jdg/1279114303Search in Google Scholar
[15] R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), no. 1, 45–76. 10.1007/BF01940959Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
