具有挤压负曲率的Cartan-Hadamard流形中的总曲率与等周不等式
Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds
AI总结：
本文针对具有挤压负曲率的Cartan-Hadamard流形，建立凸超曲面总Gauss-Kronecker曲率的尖锐下界，结合Chern-Gauss-Bonnet定理等完成证明，还应用该结果得到5维中Cartan-Hadamard猜想的等周不等式。
AI中文摘要：
我们为具有挤压负曲率的Cartan-Hadamard流形中凸超曲面的总Gauss-Kronecker曲率建立了一个尖锐下界。当直径相对于曲率尺度较小时，该下界在所有维度成立；在维度4和5中，只要挤压程度足够紧，该下界无需对直径施加任何限制即可成立。证明基于Chern-Gauss-Bonnet定理和加权Hsiung-Minkowski不等式。作为应用，我们在足够紧的曲率挤压条件下，得到了5维中Cartan-Hadamard猜想的等周不等式。
英文摘要：
We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter, provided that the pinching is sufficiently tight. The proofs are based on the Chern-Gauss-Bonnet theorem and weighted Hsiung-Minkowski inequalities. As an application, we obtain the isoperimetric inequality of the Cartan-Hadamard conjecture in dimension 5 under sufficiently pinched curvature.
