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Edward Yau Tang-wah, GBS, JP (Chinese: 邱騰華; born 28 April 1960, Hong Kong) is a Hong Kong politician. He was the Secretary for the Environment from 2007 to 2012, and from 2012 to 2017, he was the Director of the Chief Executive's Office of Hong Kong .
Shing-Tung Yau (/ jaʊ /; Chinese: 丘成桐; pinyin: Qiū Chéngtóng; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and Professor Emeritus at Harvard University.
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The Secretary for Commerce and Economic Development heads the Commerce and Economic Development Bureau of the Hong Kong Government. The Secretary is responsible for promoting commerce and the economic development of Hong Kong.
- Historical Overview
- Initial Data Sets
- The Ends of Asymptotically Flat Initial Data Sets
- Formal Statements
- Applications
- References
The original proof of the theorem for ADM mass was provided by Richard Schoen and Shing-Tung Yau in 1979 using variational methods and minimal surfaces. Edward Witten gave another proof in 1981 based on the use of spinors, inspired by positive energy theorems in the context of supergravity. An extension of the theorem for the Bondi mass was given b...
An initial data set consists of a Riemannian manifold (M, g) and a symmetric 2-tensor field k on M. One says that an initial data set (M, g, k): 1. is time-symmetric if kis zero 2. is maximal if trgk= 0 3. satisfies the dominant energy conditionif 1. 1.1. R g − | k | g 2 + ( tr g k ) 2 ≥ 2 | div g k − d ( tr g k ) | g , {\displaystyle R^{g}-|...
In the literature there are several different notions of "asymptotically flat" which are not mutually equivalent. Usually it is defined in terms of weighted Hölder spaces or weighted Sobolev spaces. However, there are some features which are common to virtually all approaches. One considers an initial data set (M, g, k) which may or may not have a ...
Schoen and Yau
Let (M, g, 0) be a time-symmetric initial data set satisfying the dominant energy condition. Suppose that (M, g) is an oriented three-dimensional smooth Riemannian manifold-with-boundary, and that each boundary component has positive mean curvature. Suppose that it has one end, and it is asymptotically Schwarzschildin the following sense: Schoen and Yau's theorem asserts that m must be nonnegative. If, in addition, the functions | x | 5 ∂ p ∂ q ∂ r h i j ( x ) , {\displaystyle |x|^{5}\partial...
Witten
Let ( M , g ) {\displaystyle (M,g)} be an oriented three-dimensional smooth complete Riemannian manifold (without boundary). Let k {\displaystyle k} be a smooth symmetric 2-tensor on M {\displaystyle M} such that 1. R g − | k | g 2 + ( tr g k ) 2 ≥ 2 | div g k − d ( tr g k ) | g . {\displaystyle R^{g}-|k|_{g}^{2}+(\operatorname {tr} _{g}k)^{2}\geq 2{\big |}\operatorname {div} ^{g}k-d(\operatorname {tr} _{g}k){\big |}_{g}.} Suppose that K ⊂ M {\displaystyle K\subset M} is an open precomp...
Extensions and remarks
According to the above statements, Witten's conclusion is stronger than Schoen and Yau's. However, a third paper by Schoen and Yau shows that their 1981 result implies Witten's, retaining only the extra assumption that | x | 4 R Φ i ∗ g {\displaystyle |x|^{4}R^{\Phi _{i}^{\ast }g}} and | x | 5 ∂ p R Φ i ∗ g {\displaystyle |x|^{5}\partial _{p}R^{\Phi _{i}^{\ast }g}} are bounded for any p . {\displaystyle p.} It also must be noted that Schoen and Yau's 1981 result relies on their 1979 result, w...
In 1984 Schoen used the positive mass theorem in his work which completed the solution of the Yamabe problem.The positive mass theorem was used in Hubert Bray's proof of the Riemannian Penrose inequality.Schoen, Richard; Yau, Shing-Tung (1979). "On the proof of the positive mass conjecture in general relativity". Communications in Mathematical Physics. 65 (1): 45–76. Bibcode:1979CMaPh..65...45S. do...Schoen, Richard; Yau, Shing-Tung (1981). "Proof of the positive mass theorem. II". Communications in Mathematical Physics. 79 (2): 231–260. Bibcode:1981CMaPh..79..231S. doi:10.1007/bf01942062. ISSN...Witten, Edward (1981). "A new proof of the positive energy theorem". Communications in Mathematical Physics. 80 (3): 381–402. Bibcode:1981CMaPh..80..381W. doi:10.1007/bf01208277. ISSN 0010-3616. S2...Ludvigsen, M; Vickers, J A G (1981-10-01). "The positivity of the Bondi mass". Journal of Physics A: Mathematical and General. 14 (10): L389–L391. Bibcode:1981JPhA...14L.389L. doi:10.1088/0305-4470...Office of the Chief Executive ( CEO) is one of the government agencies for the Hong Kong Special Administrative Region. It consists of the immediate staff to the Chief Executive of Hong Kong and multiple levels of support staff reporting to the Chief Executive. The current director is Carol Yip [ zh], making her the first woman to hold the office.
The four big families of Hong Kong (Chinese: 香港四大家族) [1][not specific enough to verify] is a term used to describe the four business families that historically rose to prominence and became influential in Hong Kong. [2] . In order of influence, they are the Li, Ho, Lo and Hui families. [2]
Nathan Road (Chinese : 彌敦道) is the main thoroughfare in Kowloon, Hong Kong, aligned south–north from Tsim Sha Tsui to Sham Shui Po. It is lined with shops and restaurants and throngs with visitors, and was known in the post– World War II years as the Golden Mile, a name that is now rarely used.
