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# Pure and Applied Mathematics Quarterly

## Volume 16 (2020)

### Number 3

### Special Issue: In Honor of Prof. Kyoji Saito’s 75th Birthday

Guest Editors: Stanislaw Janeczko, Si Li, Jie Xiao, Stephen S.T. Yau, and Huaiqing Zuo

### The isoperimetric problem in Riemannian optical geometry

Pages: 495 – 514

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n3.a6

#### Authors

#### Abstract

In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing curves subject to an area constraint are circles, and discuss implications for the photon spheres of Schwarzschild, Reissner–Nordström, as well as continuous mass models solving the Tolman–Oppenheimer–Volkoff equation. Moreover, we derive an isopermetric inequality for gravitational lensing in Riemannian optical geometry, using curve-shortening flow and the Gauss–Bonnet theorem.

#### Keywords

isoperimetric inequality, curve shortening flow, Gauss–Bonnet theorem, optical geometry, gravitational lensing

#### 2010 Mathematics Subject Classification

Primary 53C80. Secondary 83C20.

M.C.W. is grateful to Kyoji Saito for encouraging and regularly attending the *Mathematics-Astronomy Seminar Series* (2011–2014) and the *Symposium on Gravity and Light* (2013) at Kavli IPMU, University of Tokyo, which was dedicated to such mathematical aspects of gravitational lensing theory. The present collaboration was initiated at the related MRC *The Mathematics of Gravity and Light* (2018) organized by the American Mathematical Society.

Received 5 February 2019

Accepted 12 September 2019

Published 11 November 2020