We study certain natural differential forms [] and their G equivariant extensions on the space of connections. These forms are defined using the family local index theorem. When the base manifold is symplectic, they define a family of symplectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations).
We study the symplectic structure of the holomorphic coadjoint orbits, generalizing a theorem of McDuff on the symplectic structure of Hermitian symmetric spaces of noncompact type.
Gardner R J, Hug D, Weil W, et al. The Dual Orlicz-Brunn-Minkowski Theory[J]. Journal of Mathematical Analysis and Applications, 2014, 430(2): 810-829.
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Tuo Wang. The affine Pólya–Szegö principle: Equality cases and stability. 2013.
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Ye D. New Orlicz Affine Isoperimetric Inequalities[J]. Journal of Mathematical Analysis and Applications, 2014, 427(2): 905-929.
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Ahonguio F, Jossic L, Magnin A, et al. Motion and Stability of Cones in a Yield Stress Fluid[J]. Aiche Journal, 2015, 61(2): 709-717.
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Ivaki M N. The planar Busemann-Petty centroid inequality and its stability[J]. Transactions of the American Mathematical Society, 2013, 368(5): 3539-3563.
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Ambrus G, Boroczky K. Stability Results For The Volume Of Random Simplices[J]. American Journal of Mathematics, 2014, 136(4): 833-857.
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Wang T. On the Discrete Functional Lp Minkowski Problem[J]. International Mathematics Research Notices, 2015, 2015(20): 10563-10585.
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Chen F, Xu W, Yang C, et al. Rogers and Shephard inequality for the Orlicz difference body[J]. Proceedings of the American Mathematical Society, 2015, 143(9): 4029-4039.
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Fangwei Chen · Congli Yang · Miao Luo. Successive radii and Orlicz Minkowski sum. 2015.
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Van Hoang Nguyen. New approach to the affine P\'olya-Szeg\"o principle and the stability version of the affine Sobolev inequality. 2015.
We verify a conjecture of Lutwak, Yang, and Zhang about the equality case in the Orlicz-Petty projection inequality, and provide an essentially optimal stability version.
Ledrappier F, Wang X. An integral formula for the volume entropy with applications to rigidity[J]. Journal of Differential Geometry, 2009, 85(3): 461-478.
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Ni L. The Large Time Asymptotics of the Entropy[C]., 2010: 301-306.
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Itoh M, Satoh H, Suh Y J, et al. Horospheres and hyperbolicity of Hadamard manifolds[J]. Differential Geometry and Its Applications, 2014: 50-68.
Consider a compact K¨ahler manifold Mm with Ricci curvature lower bound RicM ≥ −2 (m + 1) . Assume that its universal cover
fM has maximal bottom of spectrum 1 fM = m2. Then we prove that fM is isometric to the complex hyperbolic space CHm.