@Danu @0celouvskyopoulos or: post the question in chat first, you might get better advice on where to take it

Ok

Let $M$ be a complete noncompact manifold without boundary. Suppose $M$ has nonnegative Ricci curvature outside of a ball $B_p(1)$. Suppose $f$ is harmonic in $M$ and has the property that $f=\sum_{i=1}^kv_i$, where each $v_i$ is harmonic and bounded on one side on $M-B_p(R_0)$, where $R_0>1$. We can obtain $u_i$ positive and harmonic from $v_i$ by a linear transformation. Then Yau's mean value inequality gives $$|\nabla f|(x)\le C\frac{1}{r(p,x)}\sum_{i=1}^k u_i(x).$$ [...]

From Bochner's formula we know that $|\nabla f|$ is subharmonic on $M-B_p(R_0)$ where it is $C^2$ (nonzero). Let $E$ be an end of $M$ with respect to the compact $B_p(R_0)$. Then for each $\delta>0$ the maximum principle implies $$|\nabla f|(x)-\sup_{\partial E}|\nabla f|\le C\delta\left(\sum_{i=1}^ku_i(x)+1\right)$$ for each $x\in E$.

That's the mystery inequality

I don't know how to incorporate the $\delta$ into the maximum principle argument

At best the maximum principle tells me that $|\nabla f|$ takes is supremum on $\partial E$ or at infinity.

And I don't have any growth estimate on the $u_i$'s.

So what's your question?

How does one obtain the mystery inequality

The paper and book just say "maximum principle"

I don't see what the maximum principle has to do with that $\delta$

I see

It should be possible to considerably simplify your qeustion without changin the essential meaning

It probably has to do with making $1/r$ small, but...I have no clue beyond that.

You have a tendency to get bogged down in notation and/or technical details, and that's a little bit how this feels to me too.

I don't know what's actually important here because I have zero idea of how to approach this.

So I gave the details.

I think the details are going to impact your chances of getting an answer negatively

So which details to I omit?

idk

This is definitely not in my area of expertise but it seems like not a Physics question to me. If I had to guess, I'd go for Mathematics first, and then email the author if you don't get a good response.

Yeah, definitely not PSE

Has literally nothing to do with physics :P

@Danu I will find an application for harmonic estimates in physics now

Electric fields perhaps

@SimplyBeautifulArt That one made the headlines here a while ago

@BernardoMeurer I see, then I shall delete and find the old one.

Just thought it was pretty cool

who learns probability in preschool

@0celouvskyopoulos all the kids with quantum physicist parents ofc

@blue I stand by my comment that since there are 6x as many Bengalis as Jews, you deserve a Stack (since they have one).

Can I be a moderator?

That's an interesting statistics.

@blue Please remain calm and polite at all times (this is a small warning).

yeah chillax

@blue Hola

I don't see why I should have to know Bengali

You need an outsider who doesn't have a dog in the fight

and I am a well documented friend of the Bengalese people

internet is one of the places you should not be getting serious about nothing

@BernardoMeurer I like that music video.

It's really good.

I really like it too

I once got in a fist fight in a sauna listening to that

Over new years

here's an actually good music video

looks like a boring vid

i want real crazy stuff, like this

I like this one

that one makes me ill

death grips or david bowie? :P

death grips

the bowie one is just strange

lol

yeh actually it's making me sick too

Tsc, weaklings

@BalarkaSen Do you have an idea of how to clean up my question above (search for subharmonic) for an MSE post?

I really don't know what details are important

I saw it, nope.

@defalt It has hundreds of millions of speakers, most of them dont access the Internet. For the rare fews, it would be a positive effect to their mindset. And it would be positive also for the SE. An US company, not very capable to create international systems, probably needs the experience that a nonenglish site can be useful.

@blue If you wish I would happily google a world Internet penetration map for you, although I think we can predict easily that Bangladesh won't be in the leaders on this map.

::reads through transcript:: Y'all are weird.

@ACuriousMind am I included in this insult?

Also what makes you say that

@blue What I find quite surprising here: there is a successful startup, capable to collect the above average minds on the whole world. Their obvious logical next step would be to develop their systems into a global, internationalized one. And they... seemingly intentionally give up this possibility. Why?

What's wrong with the transcript

@0celouvskyopoulos There's nothing wrong with it. It's just a weird assortment of topics :P

i can't parse what peterh is saying

maybe i'm awake for too long

@blue I talked just yesterday with a biologist from the region trying to start a Bengali site. I made a data SE query for him to find supporters. There are 10000 users only on the SO, who explicitly gave Bangladesh or Dhaka in their location.

@Balarka Sorry it is very early morning also by me

@blue The SE is doing also a lot of other things which wouldn't make me very happy being one of their share holders / investors.

i can't parse anything you are saying man. am i on drugs or what

who needs grammar anyway

i no verb

That's actually technically grammatically correct.

@blue Arabic has 200million native speakers, but the arabic SE failed recently. I think, a language site has its most visitors from the learners of this language. Thus, ell and German SE is like a rocket.

@blue But an Arabic SO had been surely a great success. All of the nonenglish SO clones are going very well. Sadly, the SE simply banned all of them, except 4. Why?..

@blue Yes, it had an initiative on a51, it were simply killed and sewed with salt. The SE simply did not want it, and also not a German or Turkish SO.

I want an Aztec language SE. Who's up for it?

@blue If they do some nasty, they clean the traces up :) you can't see deleted a51 initiatives.

die trying i guess

@blue Yeah, they have always their reasons. And the result is always that somehow you can't infect their systems with nonenglish content. They don't want a global systems, so it seems to me.

@peterh Please don't make your unfounded speculations about the motives of other people here.

@ACuriousMind I have an unfounded speculation of your motive for posting that comment. Should I say it?