Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 2

Monotone quantities of $p$-harmonic functions and their applications

Pages: 599 – 644

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n2.a1

Authors

Sven Hirsch (Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Pengzi Miao (University of Miami, Coral Gables, Florida, U.S.A.)

Luen-Fai Tam (Chinese University Hong Kong, Shatin, Hong Kong)

Abstract

We derive local and global monotonic quantities associated to $p$-harmonic functions on manifolds with nonnegative scalar curvature. As applications, we obtain inequalities relating the mass of asymptotically flat $3$-manifolds, the $p$-capacity and the Willmore functional of the boundary. As $p \to 1$, one of the results retrieves a classic relation that the ADM mass dominates the Hawking mass if the surface is area outer-minimizing.

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The first-named author’s research was partially supported by the NSF Grant No. DMS-1926686, and by the IAS School of Mathematics.

The second-named author’s research was partially supported by NSF grant DMS-1906423.

The third-named author’s research was partially supported by Hong Kong RGC General Research Fund #CUHK14301517.

Received 20 November 2022

Received revised 3 April 2023

Accepted 24 April 2023

Published 3 April 2024