论文标题
共形几何形状中的Schouten张量和非线性特征值问题的差异夹杂物
Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry
论文作者
论文摘要
令$ g_0 $为封闭的歧管$ m^n $ dimension $ n \ geq 3 $的平滑riemannian度量。我们研究了$ g_0 $的平滑度量$ g $ conformal，其schouten tensor $ a_g $满足$ m^n $上的差异包含$λ（g^{ - 1} a_g）\inγ$，其中$γ\ subset \ subset \ subset \ subset \ subset \ mathbb {r}^n $是一个标准的标准假设。这种类型的夹杂物通常在共形几何形状中的完全非线性椭圆方程的存在理论中假定。我们假设存在连续度量$ g_1 $ coldomal to $ g_0 $满足$λ（g_1^{ - 1} a_ {g_1}）\ in \ bar {γ'} $ in \ m^n $上的粘度sense in \ bar {γ'} $，与$ m^n $一起，与$ nondegenerate Elliptipity Is $ $ $ us $γ'$γ'= 2 $γ$。实际上，我们不仅证明了满足这种差异包含的指标的存在，而且还证明了Schouten Tensor的完全非线性特征值问题的存在和唯一性结果。我们还提供了结果的许多几何应用。我们表明，$σ_2$ -Yamabe问题的可溶性等同于三个维度的$σ_2$ - 操作器的非线性特征值的积极性。我们还对Aubin和Ehrlick的定理进行了概括，并在RICCI曲率的捏合中进行了概括，并在研究Green功能中应用于完全非线性的Yamabe问题。
Let $g_0$ be a smooth Riemannian metric on a closed manifold $M^n$ of dimension $n\geq 3$. We study the existence of a smooth metric $g$ conformal to $g_0$ whose Schouten tensor $A_g$ satisfies the differential inclusion $λ(g^{-1}A_g)\inΓ$ on $M^n$, where $Γ\subset\mathbb{R}^n$ is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric $g_1$ conformal to $g_0$ satisfying $λ(g_1^{-1}A_{g_1})\in\bar{Γ'}$ in the viscosity sense on $M^n$, together with a nondegenerate ellipticity condition, where $Γ' = Γ$ or $Γ'$ is a cone slightly smaller than $Γ$. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the solvability of the $σ_2$-Yamabe problem is equivalent to positivity of a nonlinear eigenvalue for the $σ_2$-operator in three dimensions. We also give a generalisation of a theorem of Aubin and Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.
