A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

关于同伦正则态射

该文在点标拓扑空间的范畴中引进了同伦正则态射的概念,研究了它存在的条件,性质以及它与同伦单（满）态,同伦正则单（满）态和同伦等价之间的密切关系。     

Classification of Isometric Immersions of the Hyperbolic Space H 2 into H 3

We transform the problem of determining isometric immersions from H ²(-1) into H ³(c) (c of solving an elliptic Monge--Ampère equation on the unit disc. Then we classify isometric immersions which possess bounded principal curvatures.     

A new explicit expression of the Contou-Carrère symbol

The aim of this note is to offer a new explicit expression of the Contou-Carrère symbol that depends only on a product of a finite number of terms. As an application, we obtain an explicit formula for a Witt Residue.

     

Analytic properties of conditional curvatures of convex hypersurfaces and the dirichlet problem for the Monge-Ampère equation

The existence and uniqueness of a surface with given geometric characteristics is one of the important topical problems of global differential geometry. By stating this problem in terms of analysis, we arrive at second-order elliptic and parabolic partial differential equations. In the present paper we consider generalized solutions of the Monge-Ampère equation ||z _ij || = ϕ(x, z, p) in Λ ⁿ , wherez = z(x ₁,...,z _n ) is a convex function,p = (p ₁,...,P _n) = (∂z/∂x ₁,...,ϖz/ϖx _n), andz _ij =ϖ ² z/ϖx _i ϖx _j. We consider the Cayley-Klein model of the space Λ ⁿ and use a method based on fixed point principle for Banach spaces. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 763–768, November, 1998.     

A complex parabolic type monge-ampère equation

The complex parabolic type Monge-Ampère equation we are dealing with is of the form inB × (0,T),u=ϕ on Γ, whereB is the unit ball in ℂ ^d ,d>1, and Γ is the parabolic boundary ofB × (0,T). Solutionu is proved unique in the class .     

A Ternary Solvent Method for Large‐Sized Two‐Dimensional Perovskites

Recent reports demonstrate that a two‐dimensional (2D) structural characteristic can endow perovskites with both remarkable photoelectric conversion efficiency and high stability, but the synthesis of ultrathin 2D perovskites with large sizes by facile solution methods is still a challenge. Reported herein is the controlled growth of 2D (C₄H₉NH₃)₂PbBr₄ perovskites by a chlorobenzene‐dimethylformide‐acetonitrile ternary solvent method. The critical factors, including solvent volume ratio, crystallization temperature, and solvent polarity on the growth dynamics were systematically studied. Under optimum reaction condition, 2D (C₄H₉NH₃)₂PbBr₄ perovskites, with the largest lateral dimension of up to 40 μm and smallest thickness down to a few nanometers, were fabricated. Furthermore, various iodine doped 2D (C₄H₉NH₃)₂PbBr_x I_4−x perovskites were accessed to tune the optical properties rationally.