各向极小曲面与周长极小化子的半空间定理
A Halfspace Theorem for Global Anisotropic Perimeter Minimizers
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中文总结 AI 辅助
该研究证明了两类各向异性半空间定理，分别针对三维偶积分元对应的各向极小曲面与任意维数的全局各向异性周长极小化子，为相关几何分析问题提供了关键理论支撑。
中文摘要 AI 辅助
我们证明了关于一致椭圆参数积分元的两个各向异性半空间定理：在三维欧氏空间ℝ³中，若偶积分元对应的连通、光滑、真嵌入且无边界的各向极小曲面位于半空间内，则其必为平面；该证明将Hoffman-Meeks论证中的悬链面替换为基于椭圆对数构造的严格外图。在任意维数下，无需假设积分元为偶，若非空边界位于半空间内的全局各向异性周长极小化子本身即为半空间；该证明结合了壁接触与平面剥离论证，此论证将任何残余相位缺陷转化为反向积分元的极小化子，并通过第二次爆破排除该缺陷。
英文摘要
We prove an arbitrary-dimensional halfspace theorem for global minimizers of a smooth uniformly elliptic anisotropic perimeter: if the nonempty boundary of a minimizing set is contained in a halfspace, then the set itself is a halfspace. No evenness of the integrand or regularity of the minimizing boundary is assumed. The proof combines wall contact with a plane-peeling argument for the complementary phase defect and a simultaneous second blow-down. We also give a direct, self-contained exterior-barrier proof of the two-dimensional stationary anisotropic statement. Its rigidity conclusion is already covered by Bergner's earlier halfspace theorem; the proof is retained because it works directly with the anisotropic Euler-Lagrange operator.
发表机构
- Hunan University（湖南大学）
机构由 AI 辅助整理，请以论文原文为准。
