u <- "ftp://ftp-projects.cen.uni-hamburg.de/seaice/AMSR2/3.125km/Arc_20120801_res3.125_pyres.nc.gz"
zf <- basename(u)
if (!file.exists(zf)) download.file(u, zf, mode = "wb")
f <- gsub(".gz$", "", zf)
if (!file.exists(f)) {
 zcon <- gzfile(zf, open = "rb")
 writeBin(readBin(zcon, "raw", n = 5e7), f)
 close(zcon)
}

library(raster)
r <- raster(f, varname = "sea_ice_concentration")
## [1] "vobjtovarid4: error #F: I could not find the requsted var (or dimvar) in the file!"
## [1] "var (or dimvar) name: Polar_Stereographic_Grid"
## [1] "file name: /home/mdsumner/amsr3/Arc_20120801_res3.125_pyres.nc"
## results in 3.125 km square pixels in the Polar Stereographic on Hughes 1980
extent(r) <- extent(-3837500, 3762500, -5362500, 5837500)
projection(r) <- "+proj=stere +lat_0=90 +lat_ts=70 +lon_0=-45 +k=1 +x_0=0 +y_0=0 +a=6378273 +b=6356889.449 +units=m +no_defs"

plot(r, col = viridis::viridis(100), zlim = c(1, 100))
library(maptools)
data(wrld_simpl)
w <- spTransform(wrld_simpl, projection(r))
plot(w, add = TRUE)

## zoom in a bit
plot(extent(c(xmin(r), xmax(r), ymin(r), ymax(r)) + c(1, -1, 1, -1) * 2e6), type = "n")
plot(r, add = TRUE, col = viridis::viridis(100), zlim = c(1, 100))
plot(w, add = TRUE)

r
## class       : RasterLayer 
## dimensions  : 3584, 2432, 8716288  (nrow, ncol, ncell)
## resolution  : 3125, 3125  (x, y)
## extent      : -3837500, 3762500, -5362500, 5837500  (xmin, xmax, ymin, ymax)
## coord. ref. : +proj=stere +lat_0=90 +lat_ts=70 +lon_0=-45 +k=1 +x_0=0 +y_0=0 +a=6378273 +b=6356889.449 +units=m +no_defs 
## data source : /home/mdsumner/amsr3/Arc_20120801_res3.125_pyres.nc 
## names       : daily.averaged.total.ice.concentration 
## z-value     : 2012-08-01 11:00:00 
## zvar        : sea_ice_concentration