关于\(\sigma_2\)-Yamabe方程的爆破现象
Blow-up phenomena for the $σ_2$-Yamabe equation
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中文总结 AI 辅助
研究针对\(n\geq27\)时\(\mathbb{S}^n\)上特定光滑度量，解决归一化\(\sigma_2\)-Yamabe方程相关问题。通过构造特定度量，克服线性化算子椭圆性可能丧失的困难，给出有限维约化中的四次轮廓确定端点维数\(n = 27\)。
中文摘要 AI 辅助
对于每个整数\(n\geq27\)，我们在\(\mathbb{S}^n\)上构造了一个在对映映射下不变且非局部共形平坦的光滑度量。对于这个固定的背景度量，归一化的\(\sigma_2\)-Yamabe方程允许一族非紧的正\(\Gamma_2^+\)-可允许解。主要困难在于线性化算子可能失去椭圆性，而标量Yamabe方程不存在此问题。我们的构造提供了保持共形度量在椭圆性锥内所需的相对衰减。有限维约化中的四次轮廓给出了端点维数\(n = 27\)。
英文摘要
For every integer $n\ge27$, we construct a smooth metric on $\mathbb S^n$ that is invariant under the antipodal map and is not locally conformally flat. For this fixed background metric, the normalized $σ_2$-Yamabe equation admits a noncompact family of positive $Γ_2^+$-admissible solutions. The main difficulty is the possible loss of ellipticity of the linearized operator. This difficulty does not occur for the scalar Yamabe equation, whose linearization has a fixed Laplace-type principal part. In the $σ_2$ problem, the positive Newton tensor of a standard bubble decays in the far field, while the terms produced by the background metric need not decay at the same rate. Our construction provides the relative decay needed to keep the conformal metrics inside the ellipticity cone. A quartic profile in the finite-dimensional reduction yields the endpoint dimension $n=27$.
