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Spacetime Harmonic Functions and Applications to Mass

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Bray, H; Hirsch, S; Kazaras, D; Khuri, M; Zhang, Y
Published in: Perspectives in Scalar Curvature
February 1, 2023

In the pioneering work of Stern, level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, are similarly effective in treating geometric inequalities involving the ADM mass. In this paper, we survey recent results in this context, focusing on applications of spacetime harmonic functions to the asymptotically flat and asymptotically hyperbolic versions of the spacetime positive mass theorem, and additionally introduce a new concept of total mass valid in both settings which is encoded in interpolation regions between generic initial data and model geometries. Furthermore, a novel and elementary proof of the positive mass theorem with charge is presented, and the level set approach to the Penrose inequality given by Huisken and Ilmanen is related to the current developments. Lastly, we discuss several open problems.

Duke Scholars

Published In

Perspectives in Scalar Curvature

Publication Date

February 1, 2023

Article type

Other

Publisher

World Scientific
 

Citation

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Bray, H., Hirsch, S., Kazaras, D., Khuri, M., & Zhang, Y. (2023). Spacetime Harmonic Functions and Applications to Mass. (M. L. Gromov & H. B. Lawson, Eds.), Perspectives in Scalar Curvature. World Scientific.
Bray, Hubert, Sven Hirsch, Demetre Kazaras, Marcus Khuri, and Yiyue Zhang. “Spacetime Harmonic Functions and Applications to Mass.” Edited by Mikhail L. Gromov and H Blaine Lawson. Perspectives in Scalar Curvature. World Scientific, February 1, 2023.
Bray H, Hirsch S, Kazaras D, Khuri M, Zhang Y. Spacetime Harmonic Functions and Applications to Mass. Gromov ML, Lawson HB, editors. Perspectives in Scalar Curvature. World Scientific; 2023.
Bray, Hubert, et al. “Spacetime Harmonic Functions and Applications to Mass.” Perspectives in Scalar Curvature, edited by Mikhail L. Gromov and H Blaine Lawson, World Scientific, 1 Feb. 2023.
Bray H, Hirsch S, Kazaras D, Khuri M, Zhang Y. Spacetime Harmonic Functions and Applications to Mass. Gromov ML, Lawson HB, editors. Perspectives in Scalar Curvature. World Scientific; 2023.

Published In

Perspectives in Scalar Curvature

Publication Date

February 1, 2023

Article type

Other

Publisher

World Scientific