AI 中文总结
该研究针对黎曼流形上静态及Ricci流演化度量下的热方程正解，基于积分对偶方法推导了带显式常数的正则化估计，为相关几何分析提供了量化支撑。
AI 中文摘要
我们研究黎曼流形上热方程的高阶全局估计，涵盖静态度量与Ricci流演化的度量两种情形。在极小几何假设下，我们推导热方程对数解的一阶正则化估计，以及带显式常数的二阶上界。该量化方法基于L. C. Evans、J.-M. Lasry和P.-L. Lions在不同情境下提出的积分对偶方法。
英文摘要
We study higher-order global estimates for the heat equation on Riemannian manifolds, both for static metrics and for metrics evolving under the Ricci flow. Under minimal geometric assumptions, we derive first-order regularizing estimates for log-solutions of the heat equation, together with upper second-order bounds with explicit constants. Our quantitative approach is based on integral duality methods proposed by L.\ C.\ Evans, J.-M.\ Lasry and P.-L.\ Lions in different settings.
