爱因斯坦-希尔伯特泛函的调和变分原理
The Harmonic Variational Principle for the Einstein-Hilbert Functional
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中文总结 AI 辅助
本文针对紧致黎曼流形的爱因斯坦-希尔伯特泛函引入受限变分原理，推导其欧拉-拉格朗日方程，证明调和临界度量等价于紧致梯度里奇孤立子的度量，并给出其规范1-形式的性质及非爱因斯坦型调和临界度量的收缩性结论。
中文摘要 AI 辅助
设(M,g)为紧致n维黎曼流形，n>2。我们对爱因斯坦-希尔伯特泛函引入受限变分原理，要求容许的度量变分满足调和规范条件。我们推导了对应的欧拉-拉格朗日方程，并证明：若一个度量相对于所有保体积的调和变分是临界的，当且仅当它的爱因斯坦张量与度量的某个倍数之差属于伴随比安基算子的像。我们证明，每个调和临界度量都确定一个紧致里奇孤立子，其孤立子常数由归一化的爱因斯坦-希尔伯特泛函给出。根据Perelman定理，每个这样的度量实际上都是紧致梯度里奇孤立子的度量。反之，每个紧致梯度里奇孤立子都满足受限欧拉-拉格朗日方程。因此，紧致黎曼度量为调和临界度量当且仅当它是紧致梯度里奇孤立子的度量。我们进一步证明，规范1-形式与孤立子势的负微分之差为基灵1-形式。特别地，若里奇张量负定，则规范1-形式消失，且该度量是爱因斯坦度量。此外，每个非爱因斯坦型的调和临界度量必然是收缩型的。
英文摘要
Let (M,g) be a compact n-dimensional Riemannian manifold, n>2. We introduce a restricted variational principle for the Einstein-Hilbert functional by requiring the admissible metric variations to satisfy the harmonic gauge condition. We derive the corresponding Euler-Lagrange equation and show that a metric is critical with respect to all volume-preserving harmonic variations if and only if its Einstein tensor differs from a multiple of the metric by an element of the image of the adjoint Bianchi operator. We prove that every harmonic critical metric determines a compact Ricci soliton whose soliton constant is given by the normalized Einstein-Hilbert functional. By Perelman's theorem, every such metric is in fact the metric of a compact gradient Ricci soliton. Conversely, every compact gradient Ricci soliton satisfies the restricted Euler-Lagrange equation. Thus, a compact Riemannian metric is harmonic critical if and only if it is the metric of a compact gradient Ricci soliton. We further show that the gauge one-form differs from the negative differential of a soliton potential by a Killing one-form. In particular, if the Ricci tensor is negative definite, then the gauge one-form vanishes and the metric is Einstein. Moreover, every non-Einstein harmonic critical metric is necessarily shrinking.
