Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Systolic $S^1$-index and characterization of non-smooth Zoll convex bodies

Stefan Matijević

TL;DR

The paper extends the Zoll condition from smooth strongly convex domains to arbitrary convex bodies by introducing the systolic $S^1$-index, defined as the Fadell–Rabinowitz index of the space of generalized systoles on the boundary. It proves that this index is a symplectic invariant and identifies a sharp characterization: a convex body is generalized Zoll if and only if the Gutt–Hutchings capacities satisfy $c^{GH}_1(K)=c^{GH}_n(K)$. The authors develop Clarke's duality to relate these invariants to spectral invariants, establish compactness and finiteness properties of the systolic spaces, and show that the space of generalized Zoll convex bodies is closed under Hausdorff convergence; they also demonstrate that interiors symplectomorphic to a ball imply generalized Zoll, with Zoll under the uniqueness of systoles. The work yields several examples, including the non-smooth case $B_\infty \times B_1$, and raises open questions about maximal indices, extremal examples, and the geometry of systoles in non-smooth settings.

Abstract

We define the systolic $S^1$-index of a convex body as the Fadell-Rabinowitz index of the space of generalized systoles associated with its boundary. We show that this index is a symplectic invariant. Using the systolic $S^1$-index, we introduce the notion of generalized Zoll convex bodies and prove that this definition coincides with the classical one when the convex body satisfies the uniqueness of systoles property, that is, when through every point passes at most one systole. Moreover, we show that generalized Zoll convex bodies can be characterized in terms of their Gutt-Hutchings capacities, and we prove that the space of generalized Zoll convex bodies is closed in the space of all convex bodies. As a corollary, we establish that if the interior of a convex body is symplectomorphic to the interior of a ball, then the convex body is generalized Zoll, and in particular Zoll if it satisfies the uniqueness of systoles property. Finally, we discuss several examples.

Systolic $S^1$-index and characterization of non-smooth Zoll convex bodies

TL;DR

The paper extends the Zoll condition from smooth strongly convex domains to arbitrary convex bodies by introducing the systolic -index, defined as the Fadell–Rabinowitz index of the space of generalized systoles on the boundary. It proves that this index is a symplectic invariant and identifies a sharp characterization: a convex body is generalized Zoll if and only if the Gutt–Hutchings capacities satisfy . The authors develop Clarke's duality to relate these invariants to spectral invariants, establish compactness and finiteness properties of the systolic spaces, and show that the space of generalized Zoll convex bodies is closed under Hausdorff convergence; they also demonstrate that interiors symplectomorphic to a ball imply generalized Zoll, with Zoll under the uniqueness of systoles. The work yields several examples, including the non-smooth case , and raises open questions about maximal indices, extremal examples, and the geometry of systoles in non-smooth settings.

Abstract

We define the systolic -index of a convex body as the Fadell-Rabinowitz index of the space of generalized systoles associated with its boundary. We show that this index is a symplectic invariant. Using the systolic -index, we introduce the notion of generalized Zoll convex bodies and prove that this definition coincides with the classical one when the convex body satisfies the uniqueness of systoles property, that is, when through every point passes at most one systole. Moreover, we show that generalized Zoll convex bodies can be characterized in terms of their Gutt-Hutchings capacities, and we prove that the space of generalized Zoll convex bodies is closed in the space of all convex bodies. As a corollary, we establish that if the interior of a convex body is symplectomorphic to the interior of a ball, then the convex body is generalized Zoll, and in particular Zoll if it satisfies the uniqueness of systoles property. Finally, we discuss several examples.
Paper Structure (10 sections, 15 theorems, 195 equations, 4 figures)

This paper contains 10 sections, 15 theorems, 195 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spaces of generalized and centralized generalized systoles
  3. On the choice of topology
  4. Relationship between spaces of generalized and centralized generalized systoles
  5. Systolic $S^1$-index of convex bodies
  6. Clarke's duality
  7. Properties of the systolic $S^1$-index
  8. Generalized Zoll convex bodies
  9. Properties of generalized Zoll convex bodies
  10. Evaluation map and generalized Zoll convex bodies

Key Result

Theorem 1.2

Let $K \subset \mathbb{R}^{2n}$ be a convex body whose interior contains the origin. Then, it holds that In particular, $\operatorname{ind}_{\operatorname{sys}}^{S^1}(K)$ is well-defined for a convex body $K \subset \mathbb{R}^{2n}$, and it holds

Figures (4)

  • Figure 1: Approximating sequence $\gamma_n$ of $\gamma$.
  • Figure 2: Systoles that do not contain the diagonals of $B_\infty$ or $B_1$.
  • Figure 3: Systoles whose projection onto $B_1$ is the horizontal diagonal of $B_1$.
  • Figure 4: A diagonal systole on $\partial(B_\infty \times B_1)$.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Proposition 2.1.1
  • ...and 27 more