Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 2

Special issue in honor of Joseph J. Kohn on the occasion of his 90th birthday

Guest Editors: J.E. Fornaess, Stanislaw Janeczko, Duong H. Phong, and Stephen S.T. Yau

A fully-nonlinear flow and quermassintegral inequalities in the sphere

Pages: 437 – 461

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n2.a4


Chuanqiang Chen (School of Mathematics and Statistics, Ningbo University, Ningbo, Zhejiang, China)

Pengfei Guan (Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada)

Junfang Li (Department of Mathematics, University of Alabama, Birmingham, Al., U.S.A.)

Julian Scheuer (School of Mathematics, Cardiff University, Cardiff, Wales, United Kingdom)


This expository paper presents the current knowledge of particular fully nonlinear curvature flows with local forcing term, so-called locally constrained curvature flows.We focus on the spherical ambient space. The flows are designed to preserve a quermassintegral and to de‑/increase the other quermassintegrals. The convergence of this flow to a round sphere would settle the full set of quermassintegral inequalities for convex domains of the sphere, but a full proof is still missing. Here we collect what is known and hope to attract wide attention to this interesting problem.

2010 Mathematics Subject Classification

Primary 35J60, 53C23. Secondary 53C42.

Chuanqiang Chen was supported by NSFC NO. 11771396.

Pengfei Guan was supported in part by NSERC Discovery Grant.

Julian Scheurer was supported in part by NSF DMS-1007223, and by the “Deutsche Forschungsgemeinschaft” (DFG, German research foundation), Project “Quermassintegral preserving local curvature flows”, No. SCHE 1879/3-1.

Received 16 February 2021

Accepted 28 June 2021

Published 13 May 2022