论文标题
具有单位体积或边界单位面积的指标的临界指标和曲率
Critical metrics and curvature of metrics with unit volume or unit area of the boundary
论文作者
论文摘要
鉴于具有边界的平滑紧凑型歧管，我们研究了体积功能和边界区域功能的变分特性，仅限于带有规定曲率的Riemannian指标的空间。我们获得了足够且必要的条件，使度量成为关键点。作为副产品，获得了非常自然的V阶段指标的类似物。在第二部分中，使用在边界环境中的Yamabe不变性，我们在带有边界的紧凑型歧管中解决了Kazdan-Warner-Kobayashi问题。在几种情况下，根据Yamabe不变的信号，我们给出了足够和必要的条件，使平滑函数成为标量或平均值的度量标准或平均曲率，对边界的体积或边界面积有限。
Given a smooth compact manifold with boundary, we study variational properties of the volume functional and of the area functional of the boundary, restricted to the space of the Riemannian metrics with prescribed curvature. We obtain a sufficient and necessary condition for a metric to be a critical point. As a by-product, a very natural analogue of V-statics metrics is obtained. In the second part, using the Yamabe invariant in the boundary setting, we solve the Kazdan-Warner-Kobayashi problem in a compact manifold with boundary. For several cases, depending on the signal of the Yamabe invariant, we give sufficient and necessary condition for a smooth function to be the scalar or mean curvature of a metric with constraint on the volume or area of the boundary.
