AI 中文总结
本文研究黎曼四流形上依赖Bach张量梯度的二次泛函，证明闭流形上存在弱Bach平行度量，给出曲面乘积分类，构造新Bach平坦度量，并建立四维柱体与三维临界度量的等价性。
AI 中文摘要
我们研究了黎曼四流形上依赖于Bach张量梯度的二次泛函的临界点，该泛函推广了Bach平坦条件。我们证明，在每个闭四流形上，存在一个弱Bach平行度量，即该泛函在共形变化下的临界点：特别地，我们证明存在无穷多个共形类，其中包含该泛函的唯一极小化子（在正常数重缩放意义下）。接下来，我们分析该泛函的全局极小值，即具有平行Bach张量的度量，并将这些度量与已知的变分问题联系起来。利用完备四流形上de Rham分裂定理的一个版本，我们借助共形梯度孤子理论，提供了曲面乘积的分类结果；我们还构造了一个新的显式Bach平坦度量，该度量既不是局部共形平坦的，也不是共形爱因斯坦的，并刻画了完备曲面上的HCMU度量。最后，我们证明了四维柱体上的Bach平行条件与三维中一个已知二次曲率泛函的临界度量存在性之间的等价性：在此方向上，我们还在某些曲率和有限能量假设下证明了平坦三维流形的刻画。
英文摘要
We study the critical points of a quadratic functional depending on the gradient of the Bach tensor on Riemannian four-manifolds, which generalize the Bach-flat condition. We show that, on every closed four-manifold, there exists a weak Bach-parallel metric, i.e. a critical point for this functional with respect to conformal variations: in particular, we prove that there exist infinitely many conformal classes which contain a unique minimizer for the functional, up to constant positive rescaling. Next, we analyze the global minima of the functional, i.e. metrics with parallel Bach tensor, relating these metrics to well-known variational problems. Using a version of de Rham's splitting theorem on complete four-manifolds, we provide a classification result for products of surfaces, exploiting the theory of conformal gradient solitons; we also construct a new explicit example of a Bach-flat metric which is neither locally conformally flat nor conformally Einstein and we characterize HCMU metrics on complete surfaces. Finally, we prove an equivalence between the Bach-parallel condition on 4D cylinders and the existence of critical metrics for a well-known quadratic curvature functional in dimension three: in this direction, we also prove a characterization of flat three-manifolds, under some curvature and finite energy assumptions.
CommentsWe fixed minor issues in some statements in the introduction and corrected some typos
