黎曼泛函的变分问题及其稳定性

The study of Riemannian functionals has long history, variational and stable problems of Riemannian functionals are important in Differential Geometry. For a closed manifold M, the character of critical point can be described by the first variational formula, and we know from the second variational formula that whether the critical metric is stable. In this program, we will consider the following problems: .(1) Study the stability of some non-Einstein metrics as critical points of a quadratic functional F(t)..(2)Kobayashi proved that the Fubini-Study metric is a minimum of the Weyl functional on 2 dimesion complex projective space, and the standard metric on S^2×S^2 is strictly stable, we will consider that whether these proposition still hold on higher dimesion. .(3) When p>2, the standard metric on sphere is a critical point of the p-th Gauss-Bonnet curvature functional, we concern with its stability. By using the first variational formula, we would find some examples of critical metrics on torus and complex projective space, and we calculate the second variational formula at the critical points, then we can judge their stability.

黎曼泛函的研究有着悠久的历史，而黎曼泛函的变分问题和稳定性问题是微分几何研究中的重要问题。对于闭流形M，一阶变分公式可以描述泛函的临界点的特征，而通过临界点处的二阶变分公式，可以进而了解该临界度量是否稳定。在本项目中，我们拟对如下的问题展开研究:.（1）研究二次泛函F(t)的非爱因斯坦度量的临界点的稳定性。.（2）Kobayashi证明了复2维射影空间上的Fubini-Study度量是Weyl泛函极小值点，以及S^2×S^2上的标准度量是Weyl泛函的严格稳定的临界点，我们拟研究在高维情形下这样的性质是否仍然成立。.（3）p>2时，球面上的标准度量是p-th Gauss-Bonnet曲率泛函的临界点，我们拟考虑该临界点的稳定性。同时利用一阶变分公式寻找环面上和复射影空间上的临界点的例子，并计算在临界点出的二阶变分，进而判断该临界点是否稳定。

黎曼泛函的研究有着悠久的历史，而黎曼泛函的变分问题和稳定性问题是微分几何研究中的重要问题。对于闭流形M，一阶变分公式可以描述泛函的临界点的特征，而通过临界点处的二阶变分公式，可以进而了解该临界度量是否稳定。在本项目中，我们对如下的问题展开了研究:.（1）Kobayashi证明了复2维射影空间上的Fubini-Study度量是Weyl泛函极小值点，我们证明了CPn上的Fubini-Study度量是严格稳定的。.（2）p>2时，球面上的标准度量是p-th Gauss-Bonnet曲率泛函的临界点，我们利用一阶变分公式寻找环面上和复射影空间上的临界点的例子，并计算了临界点出的二阶变分，进而研究了该临界点是否稳定。