AI 中文总结
该研究针对三维有界一致凸区域上可允许图的数量曲率方程，在小体积假设下证明Robin问题解的存在唯一性，得到Robin参数趋于0时的经典Neumann解，其体积阈值为仅依赖体积条件中的最优值。
AI 中文摘要
我们研究三维有界一致凸区域上可允许图的数量曲率方程的Robin问题与Neumann问题。在小体积假设下，我们证明了Robin问题解的存在性与唯一性，并得到当Robin参数趋于0时的经典Neumann解。该体积阈值在仅依赖体积的条件中是最优的。主要步骤是得到Robin参数一致的边界二阶导数估计，结合已知的内部及全局到边界的曲率估计即可得到全局界。
英文摘要
We study Robin and Neumann problems for the scalar curvature equation of admissible graphs over bounded uniformly convex domains in three dimensions. Under a small-volume assumption, we prove existence and uniqueness for the Robin problem and obtain a classical Neumann solution as the Robin parameter tends to zero. The volume threshold is optimal among conditions depending only on the volume. The main step is a boundary second-derivative estimate uniform in the Robin parameter; known interior and global-to-boundary curvature estimates then give the global bound.
