2008 Research Report
Chin-Tung Wu
I work in the area of geometric evolution equations on CR manifolds. During the past year, August 2007 - July 2008, my research concentrated on the first eigenvalues of geometric operators under CR geometric flows and the CR heat equation on CR (2n+1)-manifolds.
Conference and Workshop:
1. International Conference on Geometry and Analysis, Nanjing University, Nanjing, China, Aug. 5-11, 2007.
2. “The Harnack estimates on CR 3-manifolds ”, ICCM 2007, Zhejiang University, Zhejiang, China, Dec. 16-22, 2007.
3. “First eigenvalues of geometric operator under CR geometric flow”, AMMS 2007, Academia Sinica, Taipei, Dec. 21-24, 2007.
Advance in research:
1. We show that the first eigenvalue of -Δ_{b}+cW (c≥1/2) is nondecreasing under the Yamabe flow for closed spherical CR 3-manifolds with nonnegative Tanaka-Webster curvature and vanishing torsion.
2. We get a subgradient estimate of the CR heat equation on a closed pseudohermitian (2n+1)-manifolds. Moreover, we are able to establish Liouville-type theorems for the CR heat equation and the subgradient estimate for the CR heat kernel on (2n+1)-dimensional Heisenberg groups. (Joint with Shu-Cheng Chang & Jingzhi Tie)
pseudo-hermitian (2n+1)-manifold. Such lower bound estimates have been proved by Chiu for n = 1, by Li and Luk for n ≥ 2 and Greenleaf for n ≥ 3. (Joint with Shu-Cheng Chang)
Plans to study:
1. We will study the short time existence of the torsion flow and its propositions on a CR 3-manifold.
國 際 會 議 訪 問 報 告 書
吳 進 通此次很高興能參加在中國南京大學所辦的國際幾何與分析研討會
（International Conference on Geometry and Analysis）（會議期間：
2007 年 8 月 5 日至 11 日）。這次研討會是由著名的幾何學家田剛（Gang
Tian）教授及曹建國（Jianguo Cao）教授所主辦的，會中邀請了許多幾
何學家來報告目前在黎曼幾何、凱勒幾何、非線性偏微分方程及鏡對偁等
方面的最新進展。受邀者有：W.Ballmann, S.-C.Chang, J.-H.Cheng,K.
Fukaya,A.Futaki,K. Hirachi,J.Lott,Y.-L. Xin, J.-Y.Wu,….他們都是
