  The geometry of the neutral gas in and around galaxies is a key regulator of
the escape of ionizing photons. We present the first statistical study aiming
at linking the neutral and ionized gas distributions to the Lyman continuum
(LyC) escape fraction (fesc(LyC)) in a sample of 22 confirmed LyC leakers and
non-leakers at z~0.35 using the Keck Cosmic Web Imager (Keck/KCWI) and the Low
Resolution Spectrograph 2 (HET/LRS2). Our integral field unit data enable the
detection of neutral and low-ionization gas, as traced by Mg II, and ionized
gas, as traced by [O II], extending beyond the stellar continuum for 7 and 10
objects, respectively. All but one object with extended Mg II emission also
shows extended [O II] emission; in this case, Mg II emission is always more
extended than [O II] by a factor 1.3 on average. Most of the galaxies with
extended emission are non or weak LyC leakers (fesc(LyC) < 5%), but we find a
large diversity of neutral gas configurations around these weakly LyC-emitting
galaxies. Conversely, the strongest leakers (fesc(LyC) > 10%) appear uniformly
compact in both Mg II and [O II] with exponential scale lengths <1 kpc. We also
find a trend between fesc(LyC) and the spatial offsets of the nebular gas and
the stellar continuum emission. Moreover, we find significant anti-correlations
between the spatial extent of the neutral gas and the [O III]/[O II] ratio, and
H$\beta$ equivalent width, as well as positive correlations with metallicity
and UV size, suggesting that galaxies with more compact neutral gas sizes are
more highly ionized. The observations suggest that strong LyC emitters do not
have extended neutral gas halos and ionizing photons may be emitted in many
directions. Combined with high ionization diagnostics, we propose the Mg II,
and potentially [O II], spatial compactness are indirect indicators of LyC
emitting galaxies at high-redshift.
