Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convexity of the extended K-energy and the large time behaviour of the
  weak Calabi flow   Let $(X,\omega)$ be a compact connected K\"ahler manifold and denote by
$(\mathcal E^p,d_p)$ the metric completion of the space of K\"ahler potentials
$\mathcal H_\omega$ with respect to the $L^p$-type path length metric $d_p$.
First, we show that the natural analytic extension of the (twisted) Mabuchi
K-energy to $\mathcal E^p$ is a $d_p$-lsc functional that is convex along
finite energy geodesics. Second, following the program of J. Streets, we use
this to study the asymptotics of the weak (twisted) Calabi flow inside the
CAT(0) metric space $(\mathcal E^2,d_2)$. This flow exists for all times and
coincides with the usual smooth (twisted) Calabi flow whenever the latter
exists. We show that the weak (twisted) Calabi flow either diverges with
respect to the $d_2$-metric or it $d_1$-converges to some minimizer of the
K-energy inside $\mathcal E^2$. This gives the first concrete result about the
long time convergence of this flow on general K\"ahler manifolds, partially
confirming a conjecture of Donaldson. Finally, we investigate the possibility
of constructing destabilizing geodesic rays asymptotic to diverging weak
(twisted) Calabi trajectories, and give a result in the case when the twisting
form is K\"ahler. If the twisting form is only smooth, we reduce this problem
to a conjecture on the regularity of minimizers of the K-energy on $\mathcal
E^1$, known to hold in case of Fano manifolds.

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The categories of this paper are: hep-ex
19
The categories of this paper are: eess
17
The categories of this paper are: q-fin
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc