On Santal\'{o}'s Problem

Fang Hong

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arXiv · 2606.01771

On Santal\'{o}'s Problem

Fang Hong

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Abstract

This paper investigates an isoperimetric-type problem posed by L. A. Santal\'{o}, concerning convex surfaces in hyperbolic 3-space that minimize total mean curvature among all convex surfaces with fixed surface area. This problem asks for a characterization of the minimizers and for the optimal form of a Minkowski-type inequality in hyperbolic 3-space. In this work, we propose a conjectural description of the minimizers under certain regularity assumptions. We also construct a new family of convex surfaces as as potential minimizer candidates and establish a property of the singular points of any minimizer.

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Fang Hong. 2026-06-01. On Santal\'{o}'s Problem. https://arxiv.org/abs/2606.01771

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