There is a Batman in the Arrowverse, and he's been around for far longer than heroes like Flash and Green Arrow. It's safe to assume every live action DC show and film is somewhere in this multiverse.
The catch is that, outside of Gotham's citizens, few even believe he exists. 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. 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.
And, with Batwoman getting her own TV show on The CW, there will be plenty more Gotham City adventures in the coming years. [Discussion] Gotham's place in the Arrowverse. Dries described the look they were trying to achieve. But first, fans are curious to … 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. Baby steps, baby steps. No, they're not Arrowverse. Gotham is placed around where Milwaukee is located in Wisconsin by the Great Lakes. The show has also made it a point to make their version of Gotham distinctly different from Fox’s version. 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. 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.
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. An Arrowverse crossover with Gotham would be a whole lot trickier to pull off. RELATED: Gotham Midseason Premiere Trailer
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. However the Arrowverse applies to Earth Prime (and perhaps Earth 2), so yes they exist in this multiverse. 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. Our guys walk into a grimy, scary Gotham."
In the same interview, Stephen Amell and Grant Gustin talked about how each of their characters react to the Caped Crusader's legacy.