Patio gutter or drip edge?

You will probably have better luck with this on the home/garden board. I get that the pergola aesthetic is to leave the ends of the joists uncovered but I wouldn’t be happy with it, and neither solution is going to make the roof big enough to not have water splashing on the concrete.

That's a pretty weird design you got going on there. Seems designed for appeal rather than functionality.

A drip edge would at least stop it from running back up under the roof. I feel like a gutter would make it look even stranger.

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That's a pretty weird design you got going on there. Seems designed for appeal rather than functionality.

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Designed for both actually. The cover allows lots of light to come through, which was a major factor in the design choice. Those rear windows are the entire source of light for our main room, so a standard roof would have smothered us in darkness. We also wanted the to raise the ceiling level because our roofline would have been right over our heads giving a cramped feeling.

With that being said, there are always tradeoffs. I had anticipated the roofing cover to push the water runoff further out into the yard then it does. I think a small gutter (if feasible) could be streamlined into the look horizontally but the downspout would take away from aesthetic a bit, but a worthwhile tradeoff IMO. If I could achieve the same end goal with a drip edge however, I'm thinking that would be a more desirable solution both aesthetically and likely cost wise though I'm not real familiar with solving this kind of problem so not sure what all options exist out there. Obviously maintaining aesthetic is of value but at the end of the day, I will need to keep the water from backing onto the slab so aesthetic will have to be secondary on this one.

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We watched the LSU baseball championship game under it at midday in south LA in late June. Had the fans running and it was hot but tolerable. It seems pretty comparable to a regular patio cover in terms of heat reduction.

We opted for a thicker multi-chambered polycarbonate that is supposed to trap heat and reduce temps below.

Here's a link for reference:
LINK

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SO no hotter than say a metal or shingle and OSB roof. The project I am thinking about is the same as yours....a solid roof would block almost all natural light in the main section of the house. The website linked claims about 20 degrees cooler than direct sunlight which is significant. I am not sure steel or shingles would do that well and certainly not by much. With some air flow so 80 degrees under that canopy is soooooo much better than 100 in direct sunlight. Thanks!