发表机构
Michigan State University; Tsinghua University（密歇根州立大学; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文计算闭双曲流形上约束标量曲率下体积熵的Hessian，证明三维负定及熵上界，并给出高维符号判据与鞍点存在性。
AI 中文摘要
设$(M^n,g_0)$为闭双曲流形，$n\geq3$。我们在约束$Sc_g=-n(n-1)$下计算体积熵在$g_0$处的Hessian，并将其表示为横迹无迹对称二阶张量上的二次型。在三维情形，Hessian负定，且每个充分$C^\infty$-接近且满足$Sc_g\geq-6$的度量具有至多$2$的熵，等号恰在$g_0$的拉回时取得。在$n\geq4$维情形，一个普适谱阈值决定Hessian的符号。非零无迹Codazzi张量给出正方向，而双曲弯曲在每个这样的维度提供光滑的约束熵鞍点。
英文摘要
Let $(M^n,g_0)$ be a closed hyperbolic manifold, $n\geq3$. We compute the Hessian of volume entropy at $g_0$ under the constraint $Sc_g=-n(n-1)$ and express it as a quadratic form on transverse-traceless symmetric two-tensors. In dimension three, the Hessian is negative definite, and every sufficiently $C^\infty$-close metric with $Sc_g\geq-6$ has entropy at most $2$, with equality precisely for pullbacks of $g_0$. In dimensions $n\geq4$, a universal spectral threshold determines the Hessian sign. Nonzero trace-free Codazzi tensors give positive directions, and hyperbolic bending provides smooth constrained entropy saddles in every such dimension.
