If you want to kill your productivity, move to a new house!

I know things have been extremely slow lately, but I was moving last week (and well, that means packing everything, moving things to the new house, cleaning the old house, unpacking things at the new one… well, you get the picture).

My goal for this week is presenting Aleksandrov’s solution to  the Minkowski problem (see an earlier post I did introducing this problem). So I am going to leave you a problem as a preview, it is a sort of discrete version of the Minkowski problem:

Let   be a family of non-coplanar unit vectors in and let be positive numbers such that

Then, show that there exists a convex closed polyhedron with exactly faces with normal vectors given by and corresponding areas . Plus, this polyhedron is unique up to translation.

This is fact is not surprising (since it is not hard to check that any polyhedron has this property), but the proof is far from trivial. As you may guess, the proof cannot be constructive, it will use a continuity argument to show that there must be at least one such polyhedron. I will present this in my next post.

Books of the trade

Don’t be scared, this has not been turned into a photo blog!  Today I received my copy of Singular Integrals I ordered on-line recently, which is funny given that I have been reading that book for a long time now (at least now I won’t have to borrow a copy from the library or from a friend).

In any case, this reminded me of something I heard once: basically, that throughout the early years of your career as a mathematician, there will be a  list of books (that will depend strongly on your research interests) that you must read completely and in full detail to the point where you are be able to reproduce their contents on command. So, wondering what that list should be for me, I piled up some books from my book shelf and took a picture.

…perhaps I should be reading some of those books instead of writing this blog post

Solving the Monge-Ampere equation (continued… and finished).

I have been postponing this post for over a week due to lack of time, but finally here it is. This post ought to finish a series of past posts (here and here) where I have been describing the proof of existence of classical solutions to the Dirichlet problem for the Monge-Ampere equationvia the continuity method. Via the method of continuity  we reduced the question of existence of classical solutions to the problem of proving good a priori estimates for classical solutions, namely, we were trying to prove

Theorem (A priori estimate for the Monge-Ampere equation) Let be a smooth solution of

There is a constant (depending only on and the norm of ) such that

Last time we got almost there, using the maximum principle and the right barriers we proved the estimate

Which is still not strong enough for our needs, so in order to finish the proof of the a priori estimate, we are going to use a powerful interior estimate for concave elliptic equations, proved independently by L.C. Evans and N. Krylov in the 80’s:
Theorem (Evans-Krylov) Let be a solution of the elliptic equation

for

if is concave (or convex), then we have the following interior estimate

where depends only on , and is a universal constant.

This a well known theorem, a couple of places where one can read it are the book of Gilbarg and Trudinger (last edition), or the book of Caffarelli and Cabre. More recently, Caffarelli and Silvestre have come up with a shorter proof, still based on the original ideas of Evans and Krylov, this proof is available in arxiv. Maybe I will talk about the proof in some other post, but for now I am just going to quote the result.

The Evans-Krylov theorem is an interior result, we need also control at the boundary, that is provided by a result of Krylov

Theorem (Krylov) Let be a solution to our equation, there is a universal and constant controlled by such that  for and we have

This actually a corollary of Krylov’s theorem, which is a more general and remarkable result about equations in non-divergence form with measurable coefficients, but again I want to focus on the Monge-Ampere equation, I will talk about Krylov’s theorem some other time, a good place to read about it is the last chapter of Kazdan’s book. With these two tools its a standard argument to show that for a constant controlled by the previous two, and for , we have

I won’t do it in detail, but the proof is not too hard: basically, if the two points are closer to each other than to the boundary, then the Evans-Krylov estimate (properly scaled) gives us the inequality above, otherwise, the two points are closer to the boundary than to each other, so by the estimate of Krylov we get the same inequality in this case, and thats it!

…and that finishes the proof!. I did not present the most general result to simplify the presentation(at least for the weaker a priori estimates, which is where I did most of the details), but one can work in a more general domain (as long as it is convex) and have arbitrary boundary conditions. A much more general result, which includes not only the Monge-Ampere equation but also the Hessian equation, was proven  in a  paper by Caffarelli, Nirenberg and Spruck.

Prescribing the Ricci Curvature of Riemannian metrics

I would like to spend the next few posts talking about a problem I read about in chapter 5 of the book Einstein Manifolds by Arthur Besse (it turns out that the name Arthur Besse is made up as you can read in the preface of the book). The question that we want to address is the following

Given a compact smooth manifold without boundary , when is it possible to find a Riemannian metric in satisfying for a given Ricci candidate ?

It is of course too ambitious to try to answer this question in full generality but we can start by showing some examples of Ricci candidates for which this equation does not have a solution.

Trying to solve for amounts to solving a second order, quasilinear PDE on , however, the main difficulty here is that the operator is not elliptic.

A motivation for considering this problem comes from the question of existence of metrics with constant sectional curvature on - manifolds (compact and without boundary). This of course has to do with the celebrated theorem of Richard Hamilton on the description of - manifolds with positive Ricci curvature:

Theorem (Hamilton, 1982): Let be a connected, compact smooth dimensional manifold without boundary and suppose that admits a metric such that is positive definite everywhere. Then also admits a metric with constant sectional curvature.

We will discuss some of the ideas involved in the proof of this theorem in future posts. A consequence of this result is that is diffeomorphic to the quotient of the -sphere by a discrete group .

Back to our original problem, recall that given a Riemannian metric , the full curvature tensor is defined by

Where

Here is the Levi-Civita connection of .

The Ricci tensor is then defined as

.

Here are two basic properties of :

1) is symmetric in and , i.e.

2) In local coordinates looks as follows

We are using the summation convention (i.e we sum over repeated indices). The Christoffel symbols are defined by

Where are entries of the matrix . This says that in local coordinates we can write schematically where is a function that depends linearly on the entries of . If we then want to solve locally, property 2) tells us that we have to look at a system of the form

Property 1) tells us that an admissible Ricci candidate has to be symmetric.

One encounters obstructions for solving this system right away. One of the main difficulties has to do with the fact that the Ricci tensor satisfies the differential Bianchi identity

Where is the divergence operator respect to and is the scalar curvature of (the trace of the Ricci tensor). This says that if we define a 1-form by , then in order for to satisfy we must have

To write in coordinates, we start with

From

(which is just the definition of in coordinates) and from the expression in local coordinates of the symbols , we easily see that

As discussed in Besse’s book, Dennis DeTurck came up with examples of symmetric tensors that cannot satisfy the Bianchi identity respect to any metric. One of his examples is the following

Consider in a symmetric tensor of the form

The existence of a metric such that implies as we saw before that . In particular, from our expression for we must have

This implies that on the hyperplane , the metric must satisfy which is impossible for a Riemannian metric. It follows that for any point in the hyperplane the equation has no solution near . Notice that at these points the tensor is singular.

In the next post we will interpret the existence of examples like the one we have just discussed as a consequence of the non-ellipticity of the system .

Antonio Ache joins!

Antonio Ache, a grad student at Wisconsin will be blogging with us starting this week (after all, this is meant to be a group blog!). He is a good friend and we have known each other since our undergraduate days.

Antonio will begin by telling us about the problem of finding Riemannian metrics with given Ricci curvature. Welcome Antonio!

PS: On an unrelated note, I am writing this post from my phone! the wordpress application for the iPhone works perfectly. Yes, this means I can now blog whenever I find myself lost during a math talk…

Solving the Monge-Ampere equation (continued)

In a previous post I began the proof of the following theorem:

Let be a positive function in , then there exists a unique function such that

To prove the theorem, we looked at the function

.

we noted that it is easy to solve the problem explicitely for .  Then thanks to the inverse function theorem (the Banach space version) we saw that the set of ’s for which we can solve the equation is open (recall that was defined as the set of those in for which we can solve the equation above with right hand side  equal to ).

Since is open and non-empty, if we show that its also closed the theorem would be proved.

Part II:  Showing is closed.

(see previous post of the series for Part I)

That is closed will be shown to be a consequence of the following a priori estimate:

Theorem (A priori estimate for the Monge-Ampere equation) Let be a smooth solution of

There is a constant (depending only on and the norm of ) such that

Proving the a priori estimate is usually the hardest step, so first let’s see that the a priori estimate in fact implies that is closed: let be a sequence in converging to a number , by the a priori estimate above the sequence of solutions is uniformly bounded in , thus by the Arzela-Ascoli theorem a subsequence converges to a function , since converges to we see this is a solution of Problem , i.e. . This proves is closed and therefore the theorem.

Of course, now we “just” have to prove the a priori estimate. ..

The broad outline of its proof is: first we prove a series of a priori estimates, each stronger than the previous one, and at each given step we  use the estimates already achieved, this will bound the first and second derivatives, the main tool here is the maximum principle used as in the classical technique of Bernstein. Then, the Holder regularity of the second derivatives will be consequence of deep regularity results from the theory of fully non-linear equations, among them the Evans-Krylov theorem. Now lets go through the argument in detail, and to try to make this just a bit readable I will number the steps

Step 1) The easiest estimate is the bound

(a priori estimate (1))

where depends only on the norm of . To prove it, consider the auxiliary function , this is a smooth function that vanishes on and , picking large enough (and controlled by the supremum of ) we have

then the maximum principle implies that

in

plus is a convex function  vanishing on , so we have as well. This proves the a priori estimate (1).

Step 2) Now we go for a Lipschitz bound for , which we will refer to as the

(a priori estimate (2))

Let’s differentiate the equation in the direction of the unit vector , and we obtain

Here correspond to the entries of the cofactor matrix of , what the left hand side above amounts to is the linearization of at , which is the elliptic operator (we did this before, in the previous post). Notice also that I can talk freely about third derivatives of because any strictly convex, solution of the Monge-Ampere equation is actually (that this is true is another of the many consequences of  the Schauder theory for second order  linear elliptic equations with Holder continuous coefficients, and its not too hard to check), so the equation above holds classically. In particular, by the maximum principle (again) we conclude that for some depending on the supremum of :

but on thus there , where is the outer normal to . Here we bring back our old friend, the function defined in step 1), as we know on and they agree on the boundary, therefore

on

also, since and vanishes on we also have the inequality . Therefore the is  controlled by times a dimensional constants, i.e.  the supremum of controls . Putting the last three identities/inequalities together we get the a priori estimate (2).

Step 3) Notice that up to now we have only used the ellipticity of the equation and not the particular structure of the Monge-Ampere equation (even the differentiation step of Step 2) is a fact that holds for general elliptic equations). This  will change now that we are going to prove the

(a priori estimate (3))

We start off as in step 2 by differentiating along a fixed, arbitrary direction ,  except now we do it twice and thus we get something much more complicated than before. We get around this noting that the map is a convex function on positive definite matrices, and get an inequality that allows us to drop the lower order terms:

and this says that  is controlled by the same supremum on plus a constant controlled by .  Then we only have to estimate things on the boundary (just as for the first derivatives).

So lets pick a point and estimate the entries of . Since on we have all the first, and second order angular derivatives of at are zero, and this says a lot about the Hessian of on the boundary, its just a matter of relating partial derivatives in polar and Cartesian coordinates to see then (after doing an adequate rotation of our coordinate system) that the entries of the Hessian satisfy the relations:

for

here “” stands both for the number and the exterior normal direction since in our rotated coordinate system the vector is normal to at the point .Since we already have an a priori bound for this gives us an a priori bound for the second partial derivatives on the boundary, except for the partial derivatives of the form , .

Here we use the fact that our equation is rotation invariant (that is, the determinant is invariant under rotations of the coordinate system). Since () is diagonal, we can compute at once

I claim that there exists a   such that on , and is bounded away from zero by a quantity controlled by . Therefore there exists a  controlled by , such that

Plus, is non-negative (since is convex). Thus we have given a priori estimates for all the pure second order derivatives of on , and by the remarks made at the beginning of this step we have proved the a priori estimate (3)

Step 4) Putting the 3 previous estimates together, we have:

a priori estimate (4)

Next, we have to push this to a estimate in  . This will be the content of the next post (I am sure we can agree that this post is long enough). In the next post I  will talk about the Evans-Krylov theorem and the boundary behavior of solutions to elliptic equations. Hopefully that will wrap up this series of posts on the Dirichlet problem for Monge-Ampere,  after this I might go back to the Minkowski problem.

Quick post: A theorem and an open problem about Gauss curvature

While I finish my next post on the Dirichlet problem for the Monge-Ampere equation, I thought I could mention two neat things I have learned from reading Jerry Kazdan’s survey on Prescribing the curvature of a Riemannian manifold, a short book that I recommend strongly (its a bit out of date, but hey!, geometry has advanced quite a bit in the last couple of years!)

First, a cute solution to an easily-stated problem, as  observed by Wallach and Warner in 1970

Theorem:  Given a compact smooth 2-d manifold and is a 2-form such that , then there exists a smooth Riemannian metric on such that , where is the Gauss curvature of and is the area form of .

Proof: Pick your favorite metric on , we will  prove that the metric we are looking for is in fact conformal to , for any smooth function , define . Using the well known formulas for the Gauss curvature and area form of , one arrives at the identity

where the sub-indexes and refer to the object corresponding to the metric and , in particular, is the Laplace operator for . Then one wants to pick such that

or

but, using the metric we can write for some (or equivalently, using the Hodge star operator given by ) thus the we want is the solution of the equation

but since we have that thus by standard elliptic theory (or Fredholm theory, etc) there exists a smooth function solving equation (**). Thus the metric is the one we were looking for and the theorem is proved.

Now, an easily stated problem whose solution is not likely to be as short, and has led to a lot of research in the last 2 decades. I  read about it for the first time in an interview with Louis Nirenberg, which can be found here.

Open problem:  Given a two dimensional Riemannian manifold, can we embed it isometrically (even just locally) in three dimensional Euclidean space?

Recall that Gauss curvature solely determines the local geometry of a 2d manifold, so its not surprising that this problem is equivalent to the following problem involving the  (…tataaaaa! )  Monge-Ampere equation:

Open problem restated: Given a function , find (even just locally) a function which solves the equation

but mind you, this is not your grandpa’s Monge Ampere equation, for it is not an elliptic equation unless is strictly positive. For general , it is an equation that varies  between hyperbolic or elliptic according to the sign of , so you run into real problems in the set of points where vanishes .  If on the other hand is strictly negative, Kazdan says that the equation above is solved using tools from non-linear hyperbolic equations, of which sadly I know nothing. The case is then dealt with the techniques from the last two and the next post, which I should finish in a day or two.

Solving the Monge-Ampere equation

In my previous post, while talking about the Minkowski problem, I introduced the Monge-Ampere equation. I would like to talk about it for a post or two before continuing with the Minkowski problem, in part because the Dirichlet problem for Monge-Ampere is arguably an easier problem than the Minkowski problem.

To make the presentation cleaner, allow me to work only in the unit ball in Euclidean space, this situation contains all of the important issues one deals with to solve the equation, and it will let us avoid some of the technicalities that give little insight. In any case, all of the arguments can be tweaked without too much effort (but perhaps losing some clarity of exposition) to make them work in any smooth, strictly convex domain , the same remark goes for dealing with non-zero boundary data,  I will make more detailed remarks about this in the next post. Here we go then.

The Dirichlet problem for the Monge-Ampere equation: Consider a strictly convex domain , and a smooth, positive function . Then find a convex function twice differentiable in such that:

That a solution exists can be seen via the method of continuity (although, as I pointed out, there are other approaches). The principle behind this method is that if one can show that the set of those positive for which there is a solution is  open, closed and not empty then by connectivity there is a solution for any such positive . This is a very simple and universal method that has proven to be succesful in many situations in Mathematics (e.g. in many applications of PDEs to differential geometry), but in these post I will talk about its application to the Monge-Ampere equation.

From now on we fix , and define

this is a smooth function for every , and strictly positive for every . Thus we can consider the Dirichlet problem above with as the right hand side, lets call it “Problem “. Problem is the actual problem we want to solve, and Problem   we can solve explicitely, the solution is given by which satisfies:

Consider the set

the example above shows that . Then the strategy is to show that is at the same time open and closed in , which would imply the Dirichlet problem has a solution for any . Here is where the real work begins, and we divide it in two parts.

Part I: Showing is open.

This will be dealt with machinery from classical analysis, namely the inverse and implicit function theorems for Banach spaces. Let us consider the map

given by , this is a continuous but non-linear map between these two Banach spaces, we will apply the inverse function theorem to this map. Suppose and let be the solution to the correspoding problem, then we can assume without loss of generality that and that it is a strictly convex function. Then the differential to at can be computed by the chain rule, and it turns out that it is just the second order differential operator

where are the entries of the cofactor matrix of .  The fact that is strictly convex means that is an elliptic operator, moreover it’s coefficients are .Now I claim that is a one to one bounded linear map* from to .

This means that the differential of at  is regular, and by the inverse function theorem is a diffeomorphism between a neighborhood of in and a neighborhood of in . But then we can find such that any such that the function lies in this neighborhood, and thus one can find a such that ,  i.e. Problem has a solution and thus , which shows is open.

*Remark: The fact that is one to one is a consequence of the Schauder theory for linear 2nd order equations, but that is something that could be discussed by itself in several posts! The theory says explicitely that given a differential operator as , strictly elliptic and with coefficients, then for any there exists a unique that vanishes on and such that

That takes care of Part I. Showing that is closed is usually the more interesting part and it changes from problem to problem (showing is open is always more or less the same). I will do the second part in the next post

The Minkowski problem and the Monge-Ampere equation

So as it has been already noticed I haven’t posted much at all in a couple of months,  Sean has been doing a good job with the posting! Now that is summer and I have more free time I can get back to blogging. I won’t continue with my previous posts on minimal surfaces (at least for now), but rather talk for a couple of posts about something less heavy than geometric measure theory.

I would like to talk about the Minkowski problem. It is not only an  interesting and historically important problem in geometry but it also has deep connections with the theory of non linear elliptic equations.

The Minkowski problem goes as follows:

Given a strictly positive real function defined on , find a strictly convex compact surface such that the Gauss curvature of at the point equals , where denotes the normal to at .

The simplest example would be when is identically equal to the constant , in that case a solution to the Minkowski problem would be the sphere of radius (it also turns out that the Minkowski problem has a unique solution for a given , so the sphere is THE solution to the Minkowski problem for ).

It is my understanding (although I only know very little about the historical background) that this problem was studied intensively (among many, many other people) by A.D. Alexksandrov and  A. Pogorelov. Aleksandrov developed a theory of weak solutions and showed existence of weak solutions. Years later, via the work of many people (particularly people working in non-linear PDE) it was shown that these weak solutions were actually smooth, classical solutions, thus proving that the problem always has a unique solution.

How do PDE’s get in the picture? I will do a computation that will make this matter clear, but first let me point out the analogy with the theory of minimal surfaces and the prescription of the mean curvature of a surface: Minimal surfaces are surfaces for which the mean curvature vanishes, thus if the graph of a function happens to be a minimal surface this function solves a PDE (“the minimal surface equation”), which happens to be a quasilinear PDE. The problem of showing the existence of classical solutions for the minimal surface equation and other problems from the calculus of variations motivated much of the work in PDE through part of the 20th century (read for instance, about the work of Ennio De Giorgi on Hilbert’s 19th problem). For the Minkowski problem what we are prescribing is not the average of the principal curvatures of the surface (mean curvature) but the product of the principal curvatures (Gauss curvature), the corresponding PDE is not a quasilinear PDE, but a fully non linear PDE closely related to the Monge-Ampere equation.

To explain this further, allow me to recall a classical computation for the Gauss curvature.

Let us consider a portion of a surface given by the graph of a function u(x,y) (any small patch can be written this way if we rotate and translate our coordinate system), it is well known that one can express the Gauss curvature of this surface in terms of partial derivatives of . I am going to assume that if you are reading this you are familiar with geometry of surfaces, if not, there are a couple of good wikipedia articles on this, such as the one on the Weingarten map (aka the Shape operator).

To compute the the Gauss curvature, I am going to use the fact that the Gauss curvature equals the determinant of the differential of the Gauss map. The Gauss map is the map that associates to each point on the surface the unique unit vector parallel to the unit normal at that point), this defines a map from  the surface to the sphere . Then its differential its just a map from the tangent plane at a point in to a tangent plane to .

In the case when the surface is the graph of , in the coordinates the Gauss map is given by

Then the Weingarten map (or second fundamental form, or shape operator) its the differential of this map, its determinant is the Gauss curvature, and computing the differential (I am cheating and won’t write the differential here) we see that the Gauss curvature equals

Then, if the surface parametrizes is a solution of the Minkowski problem, then

where is the obvious function. Thus we have arrived at a PDE for

this is an equation of Monge-Ampere type.

There are several methods to construct solutions to these equations (weak solutions, approximation via polygonal surfaces, continuity method), some of these were developed independently in works of some people like A. Alexandroff, A. Pogorelov, P.L. Lions and S.Y. Cheng and S.T, Yau. One of these methods consists in building first solutions in a “weak” sense, known as Alexandroff solutions, which a priori might not be twice differentiable, so they are not solutions of the equation in the classical sense, THEN one proves that these weak solutions must actually be smooth (this is is usually called a regularity theory, and its considered the hardest part).  I will talk a bit about the existence of classical solutions on a future post on the continuity method.

Software for Pretty Diagrams

I wanted to make some pretty pictures for posts on this blog, and so I had to decide how to go about making them.  I like the hand drawn look, but it’s also nice to be able to edit using the computer.  In the end, I settled on scanning hand-drawn images, editing them in Gimp a bit, and converting them to SVG to edit more in Inkscape.

Since this is a lot of work, I have tried to automate it to the extend possible.  The result is a tiny program which works, somewhat, sometimes.  It makes diagrams that look a bit like whiteboard drawings, so I’m calling it whiteboard.  It’s badly written, badly documented, and it might set your computer on fire.  If that doesn’t scare you away, please use it for whatever you like, and if you do something cool I’d love to hear about it.