具有常$Q$-曲率、常$T$-曲率且边界极小的单位球上的共形度量
Conformal Metrics on the unit Ball with Constant $Q$-Curvature, Constant $T$-Curvature, and Minimal Boundary
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中文总结 AI 辅助
该研究完全分类了n≥4时单位球上具正常数Q、T曲率且边界极小的共形度量，发现正T曲率下存在新泡型剖面，是首个内外均含非线性项的四阶边值问题分类结果。
中文摘要 AI 辅助
我们完全分类了$n\geq4$时单位球$(\mathbb{B}^{n+1},|\mathrm{d}x|^2)$上具有正常数$Q$-曲率、正常数$T$-曲率且边界极小的共形度量。在对$Q$-曲率归一化后，对于$[0,+\infty)$内的每个$T$-曲率值，在共形微分同胚意义下存在唯一的共形度量。当$T$-曲率为正时，这些度量不是Einstein度量，并且产生了一类新的泡型剖面，除了$T=0$的情形外，均不同于Aubin--Talenti泡族。这一新现象在二阶边界Yamabe问题和闭流形上的常$Q$-曲率问题中均不存在类似情况。据我们所知，这是针对内部和边界均含非线性项的四阶边值问题的首个分类结果。
英文摘要
We completely classify conformal metrics on the unit ball $(\mathbb{B}^{n+1},|\mathrm{d} x|^2)$, $n\geq4$, with positive constant $Q$-curvature, positive constant $T$-curvature, and minimal boundary. After normalizing the $Q$-curvature, there is a unique conformal metric for each $T$-curvature value in $[0,+\infty)$, up to conformal diffeomorphism. For positive $T$-curvature, these metrics are not Einstein and yield a new family of bubble profiles, distinct from the Aubin--Talenti bubble family except when $T=0$. This new phenomenon has no analogue in either the second-order boundary Yamabe problem or the constant $Q$-curvature problem on closed manifolds. To our knowledge, this is the first classification result for a fourth-order boundary value problem with nonlinear terms both in the interior and on the boundary.
