Liouville type theorem about p-harmonic function and p-harmonic map with finite L ^q-energy

Abstract

This paper deals with the p-harmonic function on a complete non-compact submanifold M isometrically immersed in an (n + k)-dimensional complete Riemannian manifold $\overline M $ of non-negative (n−1)-th Ricci curvature. The Liouville type theorem about the p-harmonic map with finite L ^q-energy from complete submanifold in a partially non-negatively curved manifold to non-positively curved manifold is also obtained.

Keywords

p-Harmonic map / p-Harmonic map / Kato inequality / Index / Liouville theorem

References

[1] Bérard P.. A note on Bochner type theorems for complete manifolds. Manuscripta Math., 1990, 69(1): 261-266

[2] Han Y. B., Feng S. X.. A Liouville type theorem for p-harmonic function on minimal submanifold in Rn+m. Matematiqki Vesnik, 2013, 65(4): 494-498

[3] Pigola S., Veronelli G.. On the homotopy class of maps with finite p-energy into non-positively curved manifolds. Geom. Dedicata, 2009, 143: 109-116

[4] Pigolaa S., Rigolib M., Settia A. G.. Vanishing theorems on Riemannian manifolds. and geometric applications, Journal of Functional Analysis, 2005, 229: 424-461

[5] Pigola S., Rigoli M., Setti A. G.. Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds. Math. Z., 2008, 258: 347-362

[6] Wang Q. L.. Complete submanifolds in manifolds of partially non-negative curvature. Ann. Glob. Anal. Geom., 2010, 37: 113-124

[7] Yau S. T.. Some function theoretic properties of complete Riemannian manifolds and their application to geometry. Indiana Univ. Math. J., 1976, 25: 659-670