Remarks on topology of stable translating solitons
- 24 Downloads
We show that any complete f-stable translating soliton M admits no codimension one cycle which does not disconnect M. As a corollary, it follows that any two dimensional complete f-stable translating soliton has genus zero.
KeywordsTranslating solitons Stability Weighted harmonic forms
Mathematics Subject Classification (2010)Primary: 53C42 Secondary: 53C21
The first author would like to thank Miyuki Koiso and Reiko Miyaoka for their valuable comments and discussion on stability of translators. The second author was supported by Structural Materials for Innovation Strategic Innovation Promotion Program D72.
- 2.Carron, G.: \(L^2\) Harmonic Forms on Non Compact Manifolds. arXiv:0704.3194v1 (2007)
- 3.Dodziuk, J.:\(L^2\) harmonic forms on complete manifolds. In: Yau, S.T. (Ed.) Seminar on Differential Geometry, Annals of Mathematics Studies, vol. 102, pp. 291–302. Princeton University Press, Princeton (1982)Google Scholar
- 7.Impera, D., Rimoldi, M.: Quantitative Index Bounds for Translators Via Topology. arXiv:1804.07709v2 (2018)
- 9.Miyaoka, R.:\(L^2\) harmonic forms on non compact Manifolds. Geometry and Global Analysis, pp. 289–293. Tohoku University (1993)Google Scholar
- 14.Smith, G.: On Complete Embedded Translating Solitons of the Mean Curvature Flow that are of Finite Genus. arXiv:1501.04149v2 (2015)