极值Kähler方程的热流
Heat Flows for Extremal Kähler Metrics
摘要：
设$(M,J,Ω)$是闭的Kähler型极化复流形.设$G$是$(M,J)$的自同构群的极大紧子群.在Kähler度量的空间上，这些度量在$G$下是不变的，并且代表上同调类$Ω$，我们定义了一个流方程，其临界点是极值度量，这些极值度量最小化数量曲率的$L^2$ -范数的平方。我们证明了由该流定义的度量空间中的动力系统不具有周期轨道，并且它的唯一不动点是极值度量。我们证明了该流的局部时存在性，并得出结论：若解的寿命有限，则其曲率张量的范数的上确界必然随着时间的逼近而爆破，同时我们还证明了极值孤子只存在于非紧情形，并且数量曲率的全纯势的范围是一个区间，与选择来表示$Ω$的度量无关。在适当的几何条件下，我们得到了整体解的存在性和收敛性。
英文摘要：
Let $(M,J,Ω)$ be a closed polarized complex manifold of Kähler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of Kähler metrics that are invariant under $G$ and represent the cohomology class $Ω$, we define a flow equation whose critical points are extremal metrics, those that minimize the square of the $L^2$-norm of the scalar curvature. We prove that the dynamical system in this space of metrics defined by the said flow does not have periodic orbits, and that its only fixed points are extremal metrics. We prove local time existence of the flow, and conclude that if the lifespan of the solution is finite, then the supremum of the norm of its curvature tensor must blow-up as time approaches it. While doing this, we also prove that extremal solitons can only exist in the non-compact case, and that the range of the holomorphy potential of the scalar curvature is an interval independent of the metric chosen to represent $Ω$. We end up with some conjectures concerning the plausible existence and convergence of global solutions under suitable geometric conditions.
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