DIASTOLIC AND ISOPERIMETRIC INEQUALITIES ON SURFACES

DIASTOLIC AND ISOPERIMETRIC INEQUALITIES ON SURFACES

2 Feb 2024 | FLORENT BALACHEFF AND STÉPHANE SABOURAU
This paper proves a universal diastolic inequality for closed Riemannian surfaces, relating the diastole (defined via a minimax process on the one-cycle space) to the area of the surface. The inequality shows that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area. This diastolic inequality, based on an upper bound on Cheeger's constant, provides an effective method to find short closed geodesics on the two-sphere. It also shows that every Riemannian surface can be decomposed into two domains with the same area, whose common boundary has length bounded by the area. The paper compares various Riemannian invariants on the two-sphere, highlighting the special role of the diastole. The main result is a diastolic inequality for closed Riemannian surfaces of genus $ g \geq 0 $, stating that the diastole is bounded by a constant times $ \sqrt{g+1} $ times the square root of the area. The proof uses a combination of geometric and topological arguments, including a simplicial approximation and an inductive approach on the number of triangles and genus. The inequality is shown to hold for both orientable and non-orientable surfaces, and it is compared with other known inequalities in the literature.This paper proves a universal diastolic inequality for closed Riemannian surfaces, relating the diastole (defined via a minimax process on the one-cycle space) to the area of the surface. The inequality shows that any closed Riemannian surface can be swept out by a family of multi-loops whose lengths are bounded in terms of the area. This diastolic inequality, based on an upper bound on Cheeger's constant, provides an effective method to find short closed geodesics on the two-sphere. It also shows that every Riemannian surface can be decomposed into two domains with the same area, whose common boundary has length bounded by the area. The paper compares various Riemannian invariants on the two-sphere, highlighting the special role of the diastole. The main result is a diastolic inequality for closed Riemannian surfaces of genus $ g \geq 0 $, stating that the diastole is bounded by a constant times $ \sqrt{g+1} $ times the square root of the area. The proof uses a combination of geometric and topological arguments, including a simplicial approximation and an inductive approach on the number of triangles and genus. The inequality is shown to hold for both orientable and non-orientable surfaces, and it is compared with other known inequalities in the literature.
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