We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the boundary is umbilic, then the sum of the renormalized volume and the boundary integral is a conformal invariant. Analogous results are proved for asymptotically hyperbolic metrics in dimension four for which the second elementary symmetric function of the eigenvalues of the Schouten tensor is constant. Extensions and generalizations of these results are discussed. Finally, a general result is proved identifying the infinitesimal anomaly of the renormalized volume of an asymptotically hyperbolic metric in terms of its renormalized volume coefficients, and used to outline alternate proofs of the conformal invariance of the renormalized volume plus boundary integral.
The Yamabe problem for Gauss-Bonnet curvatures: a local result around space forms
在M. L. Labbi等人证明了在n维闭流形1<2k<n上具有常数2k-Gauss-Bonnet曲率的度量是Hilbert型泛函关于保体积共形变分的临界点.这就引出了相应的Yamabe问题：在一个闭流形上的任何度量是否共形于一个具有常数2k-Gauss-Bonnet曲率的度量？对于某些空间型的小扰动，我们用微扰方法肯定地回答了这个问题。更精确地说，如果（X，g）是一个非平坦的闭空间型，与一个圆球不等距，我们证明了g在度量空间中存在一个邻域U，使得U中的任何g'共形于一个其2k-Gauss-Bonnet曲率为常数的度量.
Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$
在亚黎曼海森堡空间$H^1$中证明了Gauss-Bonnet定理的一个版本。次黎曼距离使$H^1$成为度量空间，并与球面Hausdorff测度相合.利用这个测度，我们在曲面S的点处定义了一个高斯曲率，其中次黎曼分布横截于S的切空间。如果S的所有点都有这个性质，我们证明了一个Gauss-Bonnet公式，并为紧曲面（这是拓扑环面），我们得到$\int_S K = 0$。
Limit of Gaussian and normal curvatures of surfaces in Riemannian approximation scheme for sub-Riemannian three dimensional manifolds and Gauss-Bonnet theorem
