$$C_0$$ C 0 -positivity and a classification of closed three-dimensional CR torsion solitons
- PDF / 322,802 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 38 Downloads / 150 Views
Mathematische Zeitschrift
C0 -positivity and a classification of closed three-dimensional CR torsion solitons Huai-Dong Cao1 · Shu-Cheng Chang2 · Chih-Wei Chen3 Received: 18 September 2019 / Accepted: 3 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020, corrected publication 2020
Abstract A closed CR 3-manifold is said to have C0 -positive pseudohermitian curvature if (W + C0 T or )(X , X ) > 0 for any 0 = X ∈ T1,0 (M). We discover an obstruction for a closed CR 3-manifold to possess C0 -positive pseudohermitian curvature. We classify closed threedimensional CR Yamabe solitons according to C0 -positivity for C0 = 1 and the potential function lies in the kernel of Paneitz operator. Moreover, we show that any closed threedimensional CR torsion soliton must be the standard Sasakian space form. At last, we discuss the persistence of C0 -positivity along the CR torsion flow starting from a pseudo-Einstein contact form. Keywords CR Harnack quantity · CR torsion soliton · CR Paneitz operator Mathematics Subject Classification Primary 32V05 · 53C44; Secondary 32V20 · 53C56
1 Introduction Self-similar solutions, also known as geometric solitons, of various geometric flows have attracted lots of attentions in recent years because their close ties with singularity formations
Huai-Dong Cao: Research supported in part by a grant from the Simons Foundation (#586694 HC). Shu-Cheng Chang and Chih-Wei Chen: Research supported in part by the MOST of Taiwan.
B
Chih-Wei Chen [email protected]; [email protected] Huai-Dong Cao [email protected] Shu-Cheng Chang [email protected]
1
Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA
2
Department of Mathematics, Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei 10617, Taiwan, ROC
3
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, ROC
123
H. Cao et al.
in the flows. In particular, important progress has been made in the study of Ricci solitons, self-similar solutions to the mean curvature flow, as well as Yamabe solitons, etc. There have been several flows proposed to investigate the geometry and topology of CR manifolds, such as CR Yamabe flow, Q-curvature flow, CR Calabi flow, and CR torsion flow. Among them, CR torsion flow behaves more like the Riemannian Ricci flow. Similar to their Riemannian counterparts, for the CR Yamabe flow it is relatively easier to prove long time existence and convergence results than the CR torsion flow, but it provides less information about the local geometry. The CR torsion flow is to deform the CR contact form θ and the complex structure J by Tanaka-Webster curvature W and torsion A respectively. Namely, ∂J ∂t = 2 A J ,θ , (1.1) ∂θ ∂t = −2W θ. Note that the second equation alone is called the CR Yamabe flow, where to a certain extent J is freely changed. So the first equation is added to confine the behavior of J and the CR torsion tensor A. In the paper [11], it was shown that there exists a unique smooth s
Data Loading...
