DIASTOLIC AND ISOPERIMETRIC INEQUALITIES ON SURFACES

DIASTOLIC AND ISOPERIMETRIC INEQUALITIES ON SURFACES

2 Feb 2024 | FLORENT BALACHEFF AND STÉPHANE SABOURAU
The paper by Florent Balacheff and Stéphane Sabourau explores diastolic and isoperimetric inequalities on closed Riemannian surfaces. They prove a universal inequality relating the diastole, defined using a minimax process on the one-cycle space, to the area of the surface. Specifically, they show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded by a function of the surface's area. This inequality, which relies on an upper bound on Cheeger's constant, provides an effective method to find short closed geodesics on the two-sphere. The authors also demonstrate that every Riemannian surface can be decomposed into two domains with the same area, such that the length of their boundary is bounded by a function of the surface's area. They compare various Riemannian invariants on the two-sphere to highlight the special role of the diastole. The paper includes detailed proofs and examples to support these results, providing a comprehensive analysis of diastolic and isoperimetric inequalities in the context of Riemannian geometry.The paper by Florent Balacheff and Stéphane Sabourau explores diastolic and isoperimetric inequalities on closed Riemannian surfaces. They prove a universal inequality relating the diastole, defined using a minimax process on the one-cycle space, to the area of the surface. Specifically, they show that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded by a function of the surface's area. This inequality, which relies on an upper bound on Cheeger's constant, provides an effective method to find short closed geodesics on the two-sphere. The authors also demonstrate that every Riemannian surface can be decomposed into two domains with the same area, such that the length of their boundary is bounded by a function of the surface's area. They compare various Riemannian invariants on the two-sphere to highlight the special role of the diastole. The paper includes detailed proofs and examples to support these results, providing a comprehensive analysis of diastolic and isoperimetric inequalities in the context of Riemannian geometry.
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