The Geometry of (p,q)-Harmonic Maps

Open AccessArticle

by

Yan Wang

¹ and

Kaige Jiang

^2,*

¹

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

²

School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(17), 2827; https://doi.org/10.3390/math13172827

Submission received: 5 July 2025 / Revised: 25 August 2025 / Accepted: 30 August 2025 / Published: 2 September 2025

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Abstract

This paper studies

(p, q)

-harmonic maps by unified geometric analytic methods. First, we deduce variation formulas of the

(p, q)

-energy functional. Second, we analyze weakly conformal and horizontally conformal

(p, q)

-harmonic maps and prove Liouville results for

(p, q)

-harmonic maps under Hessian and asymptotic conditions on complete Riemannian manifolds. Finally, we define the

(p, q)

-

S S U

manifold and prove that non-constant stable

(p, q)

-harmonic maps do not exist.

Keywords: (p,q)-harmonic maps; variation formulas; Liouville results; stability

MDPI and ACS Style

Wang, Y.; Jiang, K. The Geometry of (p,q)-Harmonic Maps. Mathematics 2025, 13, 2827. https://doi.org/10.3390/math13172827

AMA Style

Wang Y, Jiang K. The Geometry of (p,q)-Harmonic Maps. Mathematics. 2025; 13(17):2827. https://doi.org/10.3390/math13172827

Chicago/Turabian Style

Wang, Yan, and Kaige Jiang. 2025. "The Geometry of (p,q)-Harmonic Maps" Mathematics 13, no. 17: 2827. https://doi.org/10.3390/math13172827

APA Style

Wang, Y., & Jiang, K. (2025). The Geometry of (p,q)-Harmonic Maps. Mathematics, 13(17), 2827. https://doi.org/10.3390/math13172827

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