论文标题
保形拉普拉斯第二特征值的变分特性
Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian
论文作者
论文摘要
令$（m^n，g）$为dimension $ n \ ge 3 $的封闭式riemannian歧管。假设$ [g] $是保形的laplacian $ l_g $至少具有两个负特征值。我们展示了（广义）度量的存在，该指标在所有共形指标上最大化了$ l_g $的第二个特征值（第一个特征值是Yamabe Metric最大化的）。我们还表明，最大度量定义了Yamabe方程的淋巴结解决方案，或者是向球体的谐波图。此外，我们构建了每种可能性的例子。
Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.
