非负曲率流形中子流形的Fenchel-Willmore不等式与Sobolev型不等式
Fenchel-Willmore and Sobolev-type inequalities for submanifolds in non-negatively curved manifolds
AI总结：
本文建立Fenchel-Willmore不等式与对数Sobolev不等式的联系，推广至带边界及完备非紧子流形，并导出尖锐Sobolev型不等式，应用于曲面拓扑。
摘要：
本文揭示了Fenchel-Willmore不等式与一个新的对数Sobolev不等式之间的新颖联系，该对数Sobolev不等式适用于浸入具有欧几里得体积增长的非负曲率流形中的平均凸子流形。基于这一联系，我们将Fenchel-Willmore不等式推广到带边界的子流形以及完备非紧子流形。此外，我们在相同背景下推导出一个尖锐的Sobolev型不等式。这些Sobolev型不等式具有多种应用，包括曲面情形下的拓扑推论。
英文摘要：
In this paper, we uncover a novel connection between the Fenchel-Willmore inequality and a new logarithmic Sobolev inequality for mean-convex submanifolds immersed in non-negatively curved manifolds with Euclidean volume growth. Building on this connection, we establish extensions of the Fenchel-Willmore inequality to submanifolds with boundary and to complete non-compact submanifolds. In addition, we derive a sharp Sobolev-type inequality for submanifolds in the same setting. These Sobolev-type inequalities admit a number of applications, including topological consequences in the surface case.
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