Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convex functions defined on metric spaces are pulled back to subharmonic
  ones by harmonic maps   If $u : \Omega\subset \mathbb{R}^d \to {\rm X}$ is a harmonic map valued in a
metric space ${\rm X}$ and ${\sf E} : {\rm X} \to \mathbb{R}$ is a convex
function, in the sense that it generates an ${\rm EVI}_0$-gradient flow, we
prove that the pullback ${\sf E} \circ u : \Omega \to \mathbb{R}$ is
subharmonic. This property was known in the smooth Riemannian manifold setting
or with curvature restrictions on ${\rm X}$, while we prove it here in full
generality. In addition, we establish generalized maximum principles, in the
sense that the $L^q$ norm of ${\sf E} \circ u$ on $\partial \Omega$ controls
the $L^p$ norm of ${\sf E} \circ u$ in $\Omega$ for some well-chosen exponents
$p \geq q$, including the case $p=q=+\infty$. In particular, our results apply
when ${\sf E}$ is a geodesically convex entropy over the Wasserstein space, and
thus settle some conjectures of Y. Brenier, "Extended Monge-Kantorovich theory"
in Optimal transportation and applications (Martina Franca, 2001), volume 1813
of Lecture Notes in Math., pages 91-121. Springer, Berlin, 2003.

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The categories of this paper are: hep-th
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
29
The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-th
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The categories of this paper are: patt-sol