Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convexity of the weighted Mabuchi functional and the uniqueness of
  weighted extremal metrics   We prove the uniqueness, up to a pull-back by an element of a suitable
subgroup of complex automorphisms, of the weighted extremal K\"ahler metrics on
a compact K\"ahler manifold introduced in our previous work. This extends a
result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal K\"ahler
case. Furthermore, we show that a weighted extremal K\"ahler metric is a global
minimum of a suitable weighted version of the modified Mabuchi energy. This
implies a suitable notion of weighted K-semistability of a K\"ahler manifold
admitting a weighted extremal K\"ahler metric.

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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex