Kahler度量空间的Tian适当性定理与Finsler几何
Tian's properness conjectures and Finsler geometry of the space of Kahler metrics
摘要：
著名的田的预言，存在的典型的Kahler度量应等价于各种概念的适当的Mabuchi的K-能量泛函。在某些情况下，这已经得到了验证，特别是在自同构群的限制性假设下。我们提供了反例的连续自同构的存在下，原来的猜想。该结构取决于另一种方法，适当性，在一个基本的方式使用的芬斯勒度量和它的等价物相对于群作用的度量完成。这种方法也使我们能够制定和证明新的最佳版本田的猜想在设置光滑和奇异的Kahler-Einstein度量，有或没有自同构，以及Kahler-Ricci孤子。此外，我们减少田的原始猜想（在没有自同构）和我们的修改它（在存在自同构）在一般情况下的常数数量曲率度量的猜想的正则性极小的K-能量Finsler度量完成。最后，我们的结果也解决了田的猜想的Moser-Trudinger不等式的Fano流形与Kahler-Einstein度量。
英文摘要：
Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal versions of Tian's conjecture in the setting of smooth and singular Kahler-Einstein metrics, with or without automorphisms, as well as for Kahler-Ricci solitons. Moreover, we reduce both Tian's original conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kahler-Einstein metrics.
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