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is the show gotham in the arrowverse

So far, iZombie, Lucifer and Gotham have remained separate, and their setting and tone are diverse enough that they'll probably remain that way.

Our guys walk into a grimy, scary Gotham." The various DC TV shows on The CW have made mention of Gotham City, Batman and Bruce Wayne over the years, but introducing a member of the Bat-family will finally make that corner of the Arrowverse more than Easter egg fodder. RELATED: Gotham Midseason Premiere Trailer

Dries described the look they were trying to achieve. Gotham is the remaining TV show that has yet to board the Arrowverse, granted that it is on a different network and it's a prequel piece.
And, with Batwoman getting her own TV show on The CW, there will be plenty more Gotham City adventures in the coming years. It's safe to assume every live action DC show and film is somewhere in this multiverse. I don't quite understand what you might mean by "sharing" but yes they are both set in cities of DC universe, Arrow in star city, Flash in central city and Gotham in, well, Gotham. An Arrowverse crossover with Gotham would be a whole lot trickier to pull off. Baby steps, baby steps. But first, fans are curious to … No, they're not Arrowverse. In the same interview, Stephen Amell and Grant Gustin talked about how each of their characters react to the Caped Crusader's legacy.

Having said that, Mazouz has a pretty nifty idea on how to bridge the gap between the Bruce Wayne pre-Batman phase and the bigger DCTV shared franchise.
However the Arrowverse applies to Earth Prime (and perhaps Earth 2), so yes they exist in this multiverse.

[Discussion] Gotham's place in the Arrowverse. However, after Constantine was cancelled by NBC, Berlanti's team got Matt Ryan to reprise his lead role on S4E05 of Arrow , meaning that Constantine retroactively joined the Arrowverse. Tonight on Arrow (The CW, 8/7c), the Arrowverse finally sets foot in the infamous Gotham, after so many years of merely alluding to the gritty metropolis and its resident caped crusader. But there's a huge difference in their timelines. Now with Fox’s Gotham completing its fifth and final season, the reins have been loosened up a little and the Arrowverse can now go to the forbidden land known as Gotham, although you still won’t see Batman. I don't know if this has been discussed in depth before, but the only time I remember Gotham as a city was ever mentioned in the Arrowverse was during a Supergirl episode by a family saved from danger in Metropolis, saying they were going to move there. The show has also made it a point to make their version of Gotham distinctly different from Fox’s version. The catch is that, outside of Gotham's citizens, few even believe he exists. There is a Batman in the Arrowverse, and he's been around for far longer than heroes like Flash and Green Arrow. Gotham is placed around where Milwaukee is located in Wisconsin by the Great Lakes.