Affine isoperimetric type inequalities for static convex domains in hyperbolic space

TL;DR

The paper extends affine isoperimetric theory to hyperbolic space by introducing hyperbolic ellipsoids via the orthogonal projection $\pi_{p_0}$ and defining the hyperbolic centroid. It develops hyperbolic analogs of Euclidean affine invariants, including the hyperbolic $L_p$-affine surface area, affine support function, and polar duality, and establishes two main families of inequalities, along with a Blaschke-Santaló type result in $\mathbb{H}^n$. The proofs hinge on translating hyperbolic problems to Euclidean ones through the projection, exploiting Euclidean affine isoperimetric inequalities, and using hyperbolic polar duality to characterize equality cases. This work advances affine differential geometry in curved space, with potential implications for geometric analysis and general relativity via the static convexity framework and projection techniques.

Abstract

In this paper, the notion of hyperbolic ellipsoids in hyperbolic space is introduced. Using a natural orthogonal projection from hyperbolic space to Euclidean space, we establish affine isoperimetric type inequalities for static convex domains in hyperbolic space. Moreover, equality of such inequalities is characterized by these hyperbolic ellipsoids.

Figures (1)

• Figure 2.1: Orthogonal projection $\pi$

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof