电话：010-82541521

电子信箱：xzhang@amss.ac.cn

研究方向：微分几何、广义相对论、非交换几何

一、正能量定理

（1）正宇宙常数

1998年科学家发现宇宙加速膨胀，揭示宇宙常数为正。2011年该发现被授予诺贝尔物理学奖。正宇宙常数正能量定理具现实重要性。

2010年合作证明了在正宇宙常数时，满足dominant energy condition的渐近de Sitter时空的宇宙体积增长率（即3维空间在4维时空中的平均曲率）≤de Sitter时空宇宙体积增长率时，正宇宙常数正能量猜想正确, 同时也证明了此情形时能量角动量之间的Kerr约束。

2012年合作构造了一批负总能量的例子，这时上述定理所需的其他条件都满足，只是宇宙体积增长率在某些区域超过了de Sitter时空宇宙体积增长率。从而证明上述宇宙体积增长率的条件是充分必要的，彻底研究清楚该问题。

（2）零宇宙常数

零宇宙常数时物理学家有三个关于总能量、总动量和总角动量之间关系的猜想，即总能量不小于总动量的正能量猜想、总能量不小于总角动量的Kerr约束以及引力波Bondi能量非负性猜想。1979-81年Schoen-Yau及Witten证明正能量猜想。1983年前后Schoen-Yau及多位物理学家分别用Schoen-Yau及Witten证明正能量猜想的思想方法给出Bondi能量非负性猜想的两种证明构想。

1999年独立证明Kerr约束。2004年独立证明类光无穷远正能量定理并于2006年应用该正能量定理以及应用Schoen-Yau原来给出的证明方法，合作完整证明Bondi能量非负性猜想。

拟局部量是广义相对论中的重要概念，测量有限区域的能量、动量等。很多著名的物理学家和数学家给出过各式各样拟局部质量的定义，然而至今还没有一个完美的定义能满足物理上的所有要求。

2009年，通过解Dirac方程局部边值问题成功将Witten的证明局部化到有限区域，独立给出零宇宙常数时拟局部的能量、动量和质量更好定义并证明相应量的正能量定理。

（3）负宇宙常数

物理学家普遍认为负宇宙常数时空有一神奇的性质，即著名的AdS/CFT对应关系，认为时空的引力效应等价于其共形边界上的共形场论效应。尽管负宇宙常数没有明显的宇宙学意义，但最近发现其和高温超导和凝聚态超流有着深刻的联系，这方面的一些实验数据和负宇宙常数广义相对论理论值有着“令人震惊”的一致（in striking agreement with measurements on some cupratus）。为高温超导研究提供了新的研究思路。数学上，负宇宙常数正能量猜想近年来也一直吸引着物理学家和数学家的关注和研究。如果负宇宙常数广义相对论真能解释高温超导现象，那么这时的能量不等式将反映高温超导的一些物理性质。

2015年合作证明最一般情形的总能量、总动量和总角动量不等式，研究清楚负宇宙常数正能量定理。

二、引力形变量子化与非蒸发量子黑洞

黑洞是引力量子化的基本研究对象之一。1975年，霍金提出了引力的半经典量子化理论，并用物理方法得出Schwarzschild黑洞在半经典量子化下最终将被蒸发。该工作影响巨大。

2008-2009年合作建立形变量子化数学上严格的微分几何理论，提出非交换量子爱因斯坦场方程并数学上严格证明平面波的形变量子化是场方程的真空精确解、以及Schwarzschild黑洞在该形变量子化下的非交换度量与时间无关、是不可蒸发的量子黑洞。

三、Spin几何

自1998年起，开始黎曼流形的超曲面Dirac算子特征值问题研究，并合作研究了带边流形Dirac算子特征值的最优下界估计，给出著名的Alexandrov定理一个简单的Spin几何证明。

代表论著： 

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).