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張曉

時間:2015-01-16  來源:文本大小:【 |  | 】  【打印

辦公室:N922室

電話:010-82541521

電子信箱:xzhang@amss.ac.cn

研究方向:微分幾何、廣義相對論、非交換幾何

主要成果:

一、正能量定理

(1)正宇宙常數

1998年科學家發現宇宙加速膨脹,揭示宇宙常數為正。2011年該發現被授予諾貝爾物理學獎。正宇宙常數正能量定理具現實重要性。

2010年合作證明了在正宇宙常數時,滿足dominant energy condition的漸近de Sitter時空的宇宙體積增長率(即3維空間在4維時空中的平均曲率)≤de Sitter時空宇宙體積增長率時,正宇宙常數正能量猜想正確, 同時也證明了此情形時能量角動量之間的Kerr約束。

2012年合作構造了一批負總能量的例子,這時上述定理所需的其他條件都滿足,只是宇宙體積增長率在某些區域超過了de Sitter時空宇宙體積增長率。從而證明上述宇宙體積增長率的條件是充分必要的,徹底研究清楚該問題。

(2)零宇宙常數

零宇宙常數時物理學家有三個關于總能量、總動量和總角動量之間關系的猜想,即總能量不小于總動量的正能量猜想、總能量不小于總角動量的Kerr約束以及引力波Bondi能量非負性猜想。1979-81年Schoen-Yau及Witten證明正能量猜想。1983年前后Schoen-Yau及多位物理學家分別用Schoen-Yau及Witten證明正能量猜想的思想方法給出Bondi能量非負性猜想的兩種證明構想。

1999年獨立證明Kerr約束。2004年獨立證明類光無窮遠正能量定理并于2006年應用該正能量定理以及應用Schoen-Yau原來給出的證明方法,合作完整證明Bondi能量非負性猜想。

擬局部量是廣義相對論中的重要概念,測量有限區域的能量、動量等。很多著名的物理學家和數學家給出過各式各樣擬局部質量的定義,然而至今還沒有一個完美的定義能滿足物理上的所有要求。

2009年,通過解Dirac方程局部邊值問題成功將Witten的證明局部化到有限區域,獨立給出零宇宙常數時擬局部的能量、動量和質量更好定義并證明相應量的正能量定理。

(3)負宇宙常數

物理學家普遍認為負宇宙常數時空有一神奇的性質,即著名的AdS/CFT對應關系,認為時空的引力效應等價于其共形邊界上的共形場論效應。盡管負宇宙常數沒有明顯的宇宙學意義,但最近發現其和高溫超導和凝聚態超流有著深刻的聯系,這方面的一些實驗數據和負宇宙常數廣義相對論理論值有著“令人震驚”的一致(in striking agreement with measurements on some cupratus)。為高溫超導研究提供了新的研究思路。數學上,負宇宙常數正能量猜想近年來也一直吸引著物理學家和數學家的關注和研究。如果負宇宙常數廣義相對論真能解釋高溫超導現象,那么這時的能量不等式將反映高溫超導的一些物理性質。

2015年合作證明最一般情形的總能量、總動量和總角動量不等式,研究清楚負宇宙常數正能量定理。

 

 

 

二、引力形變量子化與非蒸發量子黑洞

黑洞是引力量子化的基本研究對象之一。1975年,霍金提出了引力的半經典量子化理論,并用物理方法得出Schwarzschild黑洞在半經典量子化下最終將被蒸發。該工作影響巨大。

2008-2009年合作建立形變量子化數學上嚴格的微分幾何理論,提出非交換量子愛因斯坦場方程并數學上嚴格證明平面波的形變量子化是場方程的真空精確解、以及Schwarzschild黑洞在該形變量子化下的非交換度量與時間無關、是不可蒸發的量子黑洞。

 

 

三、Spin幾何

自1998年起,開始黎曼流形的超曲面Dirac算子特征值問題研究,并合作研究了帶邊流形Dirac算子特征值的最優下界估計,給出著名的Alexandrov定理一個簡單的Spin幾何證明。

 

代表論著: 

43.Yaohua Wang, NaqingXie, Xiao Zhang, The positive energy theorem for asymptoticallyanti-de Sitter spacetimes, Communications in Contemporary Mathematics, 17,1550015 (2015).

42.Zhuobin Liang, Xiao Zhang, Spacelike hypersurfaces with negative total energy in de Sitter spacetime, Journal of Mathematical Physics, 53, 022502 (2012).

41.Daguang Chen, OussamaHijazi, Xiao Zhang, The Dirac–Witten operator on pseudo-RiemannianManifolds, MathematischeZeitschrift, 271, 357–372 (2012).

40.Xiao Zhang, Deformation quantization and noncommutative black holes, Science in China A: Mathematics, vol.54,no.11, 2501–2508 (2011).

39.Huabin Ge, Mingxing Luo, Qiping Su, DingWang, Xiao Zhang, Bondi-Sachs metrics and photon rockets, General Relativity and Gravitation, 43, 2729–2742 (2011).

38.Wen Sun, Ding Wang, Naqing Xie, R. B. Zhang, Xiao Zhang, Gravitational collapse of spherically symmetric stars in noncommutative general relativity, Eur. Phys. J. C (2010) 69: 271-279.

37.R.B. Zhang, X. Zhang, Projective module description of embedded noncommutative spaces, World Scientific, Vol.22, No. 5 (2010) 507-531.

36.M. Luo, N. Xie, X. Zhang, Positive mass theorems for asymptotically de Sitter spacetimes, Nuclear Physics B, 825, 98-118 (2010).

35.X. Zhang, On a quasi-local mass, Classical and Quantum Gravity, 26, 245018 (2009) (9pp).

34.D. Wang, R.B. Zhang, X. Zhang, Exact solutions of noncommutative vacuum Einstein field equations and plane-fronted gravitational waves, The European Physical Journal C, 64, 439-444 (2009), DOI10.1140/epjc/s10052-009-1153-5.

33.D. Wang, R.B. Zhang, X. Zhang, Quantum deformations of Schwarzschild and Schwarzschild-de Sitter spacetimes, Classical and Quantum Gravity, 26, 085014 (2009) (14pp).

32.M. Chaichian, A. Tureanu, R. B. Zhang, X. Zhang, Riemannian Geometry of Noncommutative Surfaces, Journal of Mathematical Physics, 49, 073511 (2008) (26pp).

31.M. Chaichian, P. P. Kulish, A. Tureanu, R. B. Zhang, X. Zhang, Noncommutative fields and actions of twisted Poincare algebra, Journal of Mathematical Physics, 49, 042302 (2008) (16pp).

30.X. Zhang, A quasi-local mass for 2-spheres with negative Gauss curvature, Science in China Series A: Mathematics, 51, 1644-1650(2008).

29.X. Zhang, A new quasi-local mass and positivity, Acta Mathematica Sinica (English Series), 24, 881-890 (2008).

28.N. Xie, X. Zhang, Positive mass theorems for asymptotically AdS spacetimes with arbitrary cosmological constant, International Journal of Mathematics, 19, 285-302 (2008).

27.W.-l. Huang, X. Zhang, On the relation between ADM and Bondi energy-momenta III -- perturbed radiative spatial infinity, Science in China Series A: Mathematics, 50, 1316-1324 (2007).

26.W.-l. Huang, X. Zhang, On the relation between ADM and Bondi energy-momenta – radiative spatial infinity, 《Proceedings of ICCM 2007, December 17-22, Hangzhou》 (eds. S.T. Yau, etc). Higher Education Press, Beijing.

25.X. Zhang, On the relation between ADM and Bondi energy-momenta, Advances in Theoretical and Mathematical Physics, 10, 261-282 (2006).

24.W.-l. Huang, S.T. Yau, X. Zhang, Positivity of the Bondi mass in Bondi's radiating spacetimes, Rendiconti Lincei - Matematica e Applicazioni, 17, 335-349 (2006).

23.W.-l. Huang, X. Zhang, The energy-momentum and related topics in gravitational radiation,《Differential Geometry and Physics - Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics, 20-26 August 2005, Tianjin, China》, 248-255, Nankai Tracts in Mathematics, Vol. 10, World Scientific.

22.X. Zhang, Y-Z. Zhang, Axial anomaly for Eguchi-Hanson metrics with nonzero total mass,Communications in Theoretical Physics, 43, 79-80 (2005).

21.X. Zhang, The positive mass theorem near null infinity,《Proceedings of ICCM 2004, December 17-22, Hong Kong》 (eds. S.T. Yau, etc.), Higher Education Press, Beijing.

20.X. Zhang, A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds I, Communications in Mathematical Physics, 249, 529-548 (2004).

19.X. Zhang, Scalar flat metrics of Eguchi-Hanson type, Communications in Theoretical Physics, 42, 235-238 (2004).

18.X. Zhang, Remarks on the total angular momentum in general relativity, Communications in Theoretical Physics, 39, 521-524 (2003).

17.X. Zhang, Positive mass theorem for modified energy condition,《Morse Theory, Minimax Theorey and Their Applications to Nonlinear Differential Equations》 (eds. H. Brezis, etc.), 275-283, IP New Stud. Adv. Math. 1, International Press, Boston, 2003.

16.O. Hijazi, X. Zhang, The Dirac-Witten operator on spacelike hypersurfaces, Communications in Analysis and Geometry, 11, 737-750 (2003).

15.O. Hijazi, S. Montiel, X. Zhang, Conformal lower bounds for the Dirac operator of embedded hypersurfaces, Asian Journal of Mathematics, 6, 23-36 (2002).

14.X. Zhang, The positive mass theorem in general relativity,《Geometry and nonlinear partial differential equations》 (Hang Zhou, 2001), 227-233, AMS/IP Stud. Adv. Math. 29, Amer. Math. Soc., Providence, RI, 2002.

13.O. Hijazi, X. Zhang, Lower bounds for eigenvalues of the Dirac operator. Part II. The submanifold Dirac operator, Annals of Global Analysis and Geometry, 20, 163-181 (2001).

12.O. Hijazi, X. Zhang, Lower bounds for eigenvalues of the Dirac operator. Part I. The hypersurface Dirac operator, Annals of Global Analysis and Geometry, 19, 355-376 (2001).

11.O. Hijazi, S. Montiel, X. Zhang, Dirac operator on embedded hypersurfaces, Mathematical Research Letters, 8, 195-208 (2001).

10.O. Hijazi, S. Montiel, X. Zhang, Eigenvalues of the Dirac operator on manifolds with boundary, Communications in Mathematical Physics, 221, 255-265 (2001).

9.L. Zhang, X. Zhang, Remarks on Positive Mass Theorem, Communications in Mathematical Physics, 208, 663-669 (2000).

8.X. Zhang, Positive mass theorem for hypersurface in 5-dimensional Lorentzian manifolds, Communications in Analysis and Geometry, 8, 635-652 (2000).

7.X. Zhang, Rigidity of strongly asymptotic hyperbolic spin manifolds, Mathematical Research Letters,7, 719-72 (2000).

6.X. Zhang, A Remark: Lower bounds for eigenvalues of hypersurface Dirac operators, Mathematical Research Letters, 6, 465-466 (1999).

5.X. Zhang, Positive mass conjecture for 5-dimensional Lorentzian manifolds,Journal of Mathematical Physics, 40(7), 3540-3552 (1999).

4.X. Zhang, Angular momentum and positive mass theorem, Communications in Mathematical Physics, 206, 137-155 (1999).

3.X. Zhang, The heat flow and harmonic maps on a class of manifolds, Pacific Journal of Mathematics,182, 157-182 (1998).

2.X. Zhang, Lower bounds for eigenvalues of hypersurface Dirac operators, Mathematical Research Letters, 5, 199-210 (1998).

1.W.L. Chan, X. Zhang, Symmetries, conservation laws and Hamiltonian structures of the non-isospectral and variable coefficient KdV and MKdV equations, Journal of Physics A: Mathematics General, 28, 407-419 (1995).

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