Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 417-437

ON RICCI-HARMONIC METRICS

Lin Feng Wang

Nantong University, School of Science
Nantong 226007, P.R. China; wlf711178 'at' ntu.edu.cn

Abstract. In this paper we study gradient Ricci-harmonic soliton metrics and quasi Ricci-harmonic metrics (both metrics are called Ricci-harmonic metrics). We establish several formulas for these two metrics. Then we can show that any compact expanding or steady gradient Ricci-harmonic soliton metrics are trivial in the sense that f is a constant function, now the metric is harmonic Einstein. Rigid properties for the compact quasi Ricci-harmonic metric will also be proved. We derive the lower bound estimates of the scalar curvature for these two metrics in the noncompact case. Based on which we get the estimates of the growth of the potential function and the bottom of the L_f²-spectrum. Eventually, we discuss the diameter estimate on the compact case.

2010 Mathematics Subject Classification: Primary 53C21.

Key words: Gradient Ricci-harmonic soliton metric, quasi Ricci-harmonic metric, harmonic Einstein metric, scalar curvature, rigid property, volume growth, the bottom of the L_f²-spectrum, weak maximum principle at infinity, potential function.

Reference to this article: L.F. Wang: On Ricci-harmonic metrics. Ann. Acad. Sci. Fenn. Math. 41 (2016), 417-437.

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doi:10.5186/aasfm.2016.4127