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We prove that the DeligneLusztig variety associated to minimal length elements in any <i></i>-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in [S. Orlik, M. Rapoport, DeligneLusztig varieties and period domains over finite fields, arXiv: 0705.1646].

We prove that constant scalar curvature Khler metric adjacent to a fixed Khler class is unique up to isomorphism. The proof is based on the study of a fourth order evolution equation, namely, the Calabi flow, from a new geometric perspective, and on the geometry of the space of Khler metrics.

In this paper we prove an existence theorem of global smooth solutions for the Cauchy problem of a class of quasilinear hyperbolic systems with nonlinear dissipative terms under the assumption that only the <i>C</i>⁰-norm of the initial data is sufficiently small, while the <i>C</i>¹-norm of the initial data can be large. The analysis is based on <i>a priori</i> estimates, which are obtained by a generalised Lax transformation.