A note on Euler number of locally conformally Kähler manifolds

Holomorphic Euler number of K$\ddot{a}$hler manifolds with almost nonnegative Ricci curvature

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.

4维共形不变曲率能量的临界点的结构方程

Structural Equations for Critical Points of Conformally Invariant Curvature Energies in 4d

本文研究4维流形浸入欧氏空间中共形不变曲率能量的临界点，利用不变性和诺特定理将欧拉-拉格朗日方程化为有利结构，推广Rivière的二维方法至四维。

子流形的局部和全局共形不变量

Local and global conformal invariants of submanifolds

本文提出构造子流形共形不变量的方法，聚焦于共形子流形标量和自然积分，通过外蕴空间构造全局不变量，并推导出Gauss-Bonnet-Chern型公式，证明了共形紧致极小子流形的刚性结果。

A conformally invariant Yang-Mills type energy and equation on 6-manifolds

We define a conformally invariant action S on gauge connections on a closed pseudo-Riemannian manifold M of dimension 6. At leading order this is quadratic in the gauge connection. The Euler-Lagrange equations of S, with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang-Mills equations. For any gauge connection A on M, we define S(A) by first defining a Lagrangian density associated to A. This is not conformally invariant but has a conformal transformation analogous to a Q-curvature. Integrating this density provides t…

Rigidité conforme des hémisphères S^4_+ et S^6_+

Let (M,g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its Yamabe invariant is positive, then (M,g) is conformally isometric to the standard hemisphere. As an application and using a result of Hang-Wang, we prove a rigidity result for these hemispheres regarding the Min-Oo conjecture.

On compact manifolds admitting indefinite metrics with parallel Weyl tensor

Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classes, and infiniteness of the fundamental group. We also show that, in the Lorentzian case, each of them is at least 5-dimensional and admits a two-fold cover which is a bundle over the circle.