中文摘要：

设（M，g）为紧致仿射Kaehler流形，仿射Kaehler度量g=∑fijdxidxj，作者证明了若f满足Δlog（det（fij））=0及Ricci曲率半正定，则M是R^n／Г，其中Г为R^n上离散等距子群．进一步，对光滑函数h，作者考虑M上的变分问题，其Euler-Lagrange方程为Alog（det（fij））=4h（det（fij））^-1/2，通过解这个四阶方程的一类边值问题，构造了定义在R^n上的欧氏完备仿射Kaehler流形.

英文摘要：

Let （M, g） be a n dimenional compact affine Kaehler manifold, its Kaehler metric is g=∑fijdxidxj If Δlog（det（fij）） = 0 and its Ricci curvature Rij≥0, then M must be R^n/Г, where Г be a subgroup of isometric of R^n which acts freely and properly discontinuously on R^n. Moreover, for a smooth function h, a more general volume variational problem on M is considered, the Euler-Lagrange equation is Alog（det（fij ））= 4h （det（f/ij））^-1/2, by solving some boundary problem of the 4-order equation, many Euclidean complete affine Kaehler manifold are constructed.