Harmonic manifolds with minimal horospheres |
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Authors: | Akhil Ranjan Hemangi Shah |
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Institution: | (1) Department of Mathematics, Indian Institute of Technology, Powai, 400076 Mumbai, India |
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Abstract: | For a non-compact harmonic manifold M, we establish an integral formula for the derivative of a harmonic function on M. As
an application we show that for the harmonic spaces having minimal horospheres, bounded harmonic functions are constant. The
main result of this article states that the harmonic spaces having polynomial volume growth are flat. In other words, if the
volume density function Θ of M has polynomial growth, then M is flat. This partially answers a question of Szabo namely, which
density functions determine the metric of a harmonic manifold. Finally, we give some natural conditions which ensure polynomial
growth of the volume function. |
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Keywords: | Math Subject Classifications" target="_blank">Math Subject Classifications primary 53C21 secondary 53C25 |
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