AI 中文总结
该研究在紧凯勒流形上引入规定标量曲率测度方程，证明正标量曲率凯勒度量存在性等四个命题等价，还得出正维紧光滑环面凯勒流形的凯勒类必含对应不变度量，且方程有唯一模常数的光滑解。
AI 中文摘要
我们在紧凯勒流形上引入了规定标量曲率测度方程。对于总标量曲率为正的凯勒类，我们证明以下命题等价：正标量曲率凯勒度量的存在性、该方程对每个容许测度的可解性、相关泛函的d₁-强制性，以及沿有限能量d₁-测地射线的一致测地稳定性。由此可得，在每个固定凯勒类中，正标量曲率凯勒度量的空间要么为空，要么是可缩的。我们进一步证明，正维紧光滑环面凯勒流形上的每个凯勒类都包含一个环面不变的正标量曲率度量。因此，对于每个凯勒类，规定标量曲率测度方程对每个容许测度都存在唯一模常数的光滑解。
英文摘要
We introduce the prescribed scalar curvature measure equation on a compact Kähler manifold. For a Kähler class of positive total scalar curvature, we prove that the following are equivalent: the existence of a positive scalar curvature Kähler metric, solvability of this equation for every admissible measure, $d_1$-coercivity of the associated functionals, and uniform geodesic stability along finite-energy $d_1$-geodesic rays. As a consequence, in each fixed Kähler class, the space of positive scalar curvature Kähler metrics is either empty or contractible. We further prove that every Kähler class on a positive-dimensional compact smooth toric Kähler manifold contains a torus-invariant metric of positive scalar curvature. Therefore, for every Kähler class, the prescribed scalar curvature measure equation admits a smooth solution for every admissible measure, unique modulo constants.
Comments45 pages, all comments are welcome
