Details
| Original language | English |
|---|---|
| Pages (from-to) | 23-47 |
| Number of pages | 25 |
| Journal | New York journal of mathematics |
| Volume | 9 |
| Publication status | Published - 1 Dec 2003 |
| Externally published | Yes |
Abstract
If L ⊂ M is a Legendre submanifold in a Sasaki manifold, then the mean curvature flow does not preserve the Legendre condition. We define a kind of mean curvature flow for Legendre submanifolds which slightly differs from the standard one and then we prove that closed Legendre curves L in a Sasaki space form M converge to closed Legendre geodesics, if k2 + σ + 3 ≤ 0 and rot(L) = 0, where σ denotes the sectional curvature of the contact plane ξ and k and rot(L) are the curvature respectively the rotation number of L. If rot(L) ≠ 0, we obtain convergence of a subsequence to Legendre curves with constant curvature. In case σ + 3 ≤ 0 and if the Legendre angle α of the initial curve satisfies osc (α) ≤ π, then we also prove convergence to a closed Legendre geodesic.
Keywords
- Geodesic, Hamiltonian minimal, Lagrangian, Lagrangian cone, Legrendrian, Mean curvature flow, Minimal, Volume decreasing
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: New York journal of mathematics, Vol. 9, 01.12.2003, p. 23-47.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Closed Legendre geodesics in Sasaki manifolds
AU - Smoczyk, Knut
PY - 2003/12/1
Y1 - 2003/12/1
N2 - If L ⊂ M is a Legendre submanifold in a Sasaki manifold, then the mean curvature flow does not preserve the Legendre condition. We define a kind of mean curvature flow for Legendre submanifolds which slightly differs from the standard one and then we prove that closed Legendre curves L in a Sasaki space form M converge to closed Legendre geodesics, if k2 + σ + 3 ≤ 0 and rot(L) = 0, where σ denotes the sectional curvature of the contact plane ξ and k and rot(L) are the curvature respectively the rotation number of L. If rot(L) ≠ 0, we obtain convergence of a subsequence to Legendre curves with constant curvature. In case σ + 3 ≤ 0 and if the Legendre angle α of the initial curve satisfies osc (α) ≤ π, then we also prove convergence to a closed Legendre geodesic.
AB - If L ⊂ M is a Legendre submanifold in a Sasaki manifold, then the mean curvature flow does not preserve the Legendre condition. We define a kind of mean curvature flow for Legendre submanifolds which slightly differs from the standard one and then we prove that closed Legendre curves L in a Sasaki space form M converge to closed Legendre geodesics, if k2 + σ + 3 ≤ 0 and rot(L) = 0, where σ denotes the sectional curvature of the contact plane ξ and k and rot(L) are the curvature respectively the rotation number of L. If rot(L) ≠ 0, we obtain convergence of a subsequence to Legendre curves with constant curvature. In case σ + 3 ≤ 0 and if the Legendre angle α of the initial curve satisfies osc (α) ≤ π, then we also prove convergence to a closed Legendre geodesic.
KW - Geodesic
KW - Hamiltonian minimal
KW - Lagrangian
KW - Lagrangian cone
KW - Legrendrian
KW - Mean curvature flow
KW - Minimal
KW - Volume decreasing
UR - http://www.scopus.com/inward/record.url?scp=3042643078&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:3042643078
VL - 9
SP - 23
EP - 47
JO - New York journal of mathematics
JF - New York journal of mathematics
SN - 1076-9803
ER -