A Liouville theorem for certain nonstationary maps. (English) Zbl 0659.49019

Let u: (M,g)\(\to (N,h)\) be a map between Riemannian manifolds and let \(E(u)=\int_{M}| du|^ p dv(g)\), \(p\geq 2\) be the \(L^ p\) energy of u, where (M,g) is the Euclidean space \({\mathbb{R}}^ n\) or the hyperbolic space \(H^ n\), \(n\geq p\). The author shows that if E(u) grows sufficiently slowly and if its first r-variation satisfies a certain integral inequality, then u is constant almost everywhere.
Reviewer: C.Udrişte


49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53C20 Global Riemannian geometry, including pinching
58E99 Variational problems in infinite-dimensional spaces
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text: DOI


[1] Allard, W., On the first variation of a varifold, Ann. Math., 95, 417-491 (1972) · Zbl 0252.49028
[2] Price, P., A monotonicity formula for Yang-Mills fields, Manuscripta math., 43, 131-166 (1983) · Zbl 0521.58024
[3] Hardt, R.; Lin, F-H., Mappings minimizing the \(L^p\) norm of the gradient (1986), CMA: CMA Canberra, Preprint
[4] Sibner, L. M., An existence theorem for a nonregular variational problem, Manuscripta math., 43, 45-72 (1983) · Zbl 0534.58022
[5] Garber, W-D.; Ruijsenaars, S. N.M.; Seiler, E.; Burns, D., On finite action solutions of the nonlinear \(σ\)-model, Ann. Phys., 119, 305-325 (1979) · Zbl 0412.35089
[6] Hildebrandt, S., (Chern, S. S.; Wen-Tsün, Wu, Proc. Beijing Symp. diff. Geometry diff. Eqns, Beijing, 1980. Proc. Beijing Symp. diff. Geometry diff. Eqns, Beijing, 1980, Nonlinear elliptic systems and harmonic mappings, Vol. 1 (1982), Gordon & Breach: Gordon & Breach New York), 481-615 · Zbl 0515.58012
[7] Kazdan, J. L., Parabolicity and the Liouville property on complete Reimannian manifolds, (Tromba, A., Seminar on New Results in Nonlinear Partial Differential Equations (1987), Max-Planck Inst: Max-Planck Inst Bonn), 153-166 · Zbl 0626.58021
[8] Simon, L., Lectures on Geometric Measure Theory, Proc. CMA Aust. Natn. Univ., Vol. 3 (1983), Canberra · Zbl 0546.49019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.