The isoperimetric problem is a classic topic in geometric measure theory, yet critical questions regarding the characterization of optimal solutions -- even asymptotically optimal ones -- remain largely unresolved. In this paper, we investigate the large deviations asymptotics (governing sequences of exponentially small sets) for the isoperimetric problem on the product Riemannian manifold $M^{n}$ endowed with the product probability measure $ν^{\otimes n}$, where $M$ is a Riemannian manifold satisfying $\mathrm{CD}(0,\infty)$. When the probability measure $ν$ admits a finite moment generating function for squared distance, we establish an exact characterization of the large deviations asymptotics of the isoperimetric profile, which reveals a precise equivalence between this asymptotic isoperimetry and an established strengthening of log-Sobolev inequality -- namely, nonlinear log-Sobolev inequality. It is observed that conditional typical sets, a fundamental concept from information theory, form an asymptotically optimal solution to the isoperimetric problem. This class of subsets further yields an upper bound on the isoperimetric profile in the central limit regime (concerning constant-volume sets), expressed as the Gaussian isoperimetric profile times the square root of spectral gap. Furthermore, we apply our results to establish quantitative relations among optimal constants in isoperimetric, concentration and transport inequalities, additionally covering isoperimetry for subexponentially and superexponentially small sets. Our results provide a rigorous justification from the perspective of nonlinear log-Sobolev inequalities for why isoperimetric minimizers behave fundamentally differently across spaces with distinct geometric structures. Our proof idea integrates tools from information theory, optimal transport, and geometric measure theory.

翻译：等周问题是几何测度论中的经典课题，然而关于最优解（甚至渐近最优解）刻画的关键问题在很大程度上仍未解决。本文研究了乘积黎曼流形 $M^{n}$ 上等周问题的大偏差渐近性（控制指数级小集合序列），其中 $M$ 是满足 $\mathrm{CD}(0,\infty)$ 条件的黎曼流形，并赋予乘积概率测度 $ν^{\otimes n}$。当概率测度 $ν$ 对平方距离具有有限矩生成函数时，我们建立了等周剖面大偏差渐近性的精确刻画，揭示了这种渐近等周性与对数索博列夫不等式的一种已知强化形式——即非线性对数索博列夫不等式——之间的精确等价关系。研究发现，信息论中的基本概念——条件典型集——构成了等周问题的渐近最优解。这类子集进一步给出了中心极限区域（涉及恒定体积集合）中等周剖面的上界，其表达式为高斯等周剖面乘以谱隙的平方根。此外，我们应用所得结果建立了等周不等式、集中不等式和传输不等式中最优常数之间的定量关系，同时涵盖了次指数级和超指数级小集合的等周性。我们的研究结果为等周极小元在不同几何结构的空间中表现出根本性差异这一现象，提供了来自非线性对数索博列夫不等式视角的严格理论依据。证明思路融合了信息论、最优传输和几何测度论的工具。