Summaries and comments on some papers

Below are explanations about the content of some of my papers. They are meant to present the main ideas in an informal way.

We consider a simple hamiltonian on n particles in d-dimensional Euclidean space, given by a Riesz (i.e. inverse power) repulsive interaction (or a logarithmic repulsive interaction if d=1,2) between distinct particles, plus a well-rescaled interaction with a potential, which grows fast enough at infinity so that the particles are confined and unable to escape to infinity. The pair interaction is with a negative power in [d-2,d[ (or logarithmic in dimensions 1,2), the case of a negative -(d-2) power corresponding to the Coulomb potentials.

The question which we address is how to characterize and explain the coefficients of the power expansion (in n) of the equilibrium energy.

The leading order term has a square power of n and has as a coefficient the Gamma-limit of our energy, which is a functional on probability measures, and is relatively well understood in good generality, starting with the works of Frostman and of Choquet.

The main focus here is on the next term in the expansion. This term consists of a functional interpretable as an energy on microscopic configurations of our interacting points as n becomes large, renormalized with a -(1+s/d) power of n. The rescalements of such configurations consist of infinitely many points which are relatively evenly distributed in the whole space, and are interpretable as positive point charges interacting (via our Riesz interaction) with a neutralizing negative uniform background charge.

It is conjectured that special lattices are the minimizers of the microscopic energy, at least in dimensions 2,4,8,24. In particular the famous Abrikosov triangular lattice is also observed in superconductors, in dimension 2.

While the conjecture remains wide open, we prove that among 2D lattices the microscopic energy is minimized by the Abrikosov lattice even in the case of Riesz interactions. To obtain this we express our minimization in terms of the Fourier transforms of the potentials which arise, and by controlling the convergence of the Fourier series we recover the analytic continuation of the Epstein zeta function, which was already known to be minimized by the triangular lattice by works of Cassels, Rankin, Ennola, Dianada.

This work extends and retrieves previous results available in the case of Coulomb interactions, to Riesz type interactions in a range where the potentials can be interpreted in terms of the fractional Laplacian. Except for the above link to the zeta function which is based on Fourier theory, the study is based on the idea present in Caffarelli-Silvestre of reinterpreting the (non-local) fractional Laplacian as a local operator by adding an extra dimension to the space, with an appropriate weight.

Some Motivations

The kind of model studied here arises from several different motivations, including random matrix ensembles in dimensions 1,2 and approximation questions, e.g. the theory of Fekete points. The special case of Coulomb type interactions appears in the study of thin superconductors immersed in strong magnetic fields.

The situation here can be also loosely interpreted as a toy equilibrium model of (linear) interactions of topological singular points like those appearing in the theory of harmonic maps or of singular energy-minimizing bundles. Note that in that case unlike here the interaction between singularities is highly nonlinear, in other words the error to the superposition principle becomes uncontrolled as the singularities condensate. Therefore a similar precise description of the expansion for the equilibrium energy as a series in the number of singular points is not yet available.

In this paper I prove that it is possible to find minimizers for the Yang-Mills energy in 5 dimensions which have singularities. In a preceding paper we proved together with Tristan Rivière that minimizers have isolated singular points, therefore this result proves the optimality of this partial regularity.

The example I find here is the "model case", i.e. I fix the most symmetric SU(n) connection on the sphere S⁴, and extend it over B⁵ \ {0} by pulling back via radial projection. This is analogous to the result by Hardt-Lin-Poon that x/|x| on B³ is a minimizer for the Dirichlet energy among maps in W^{1,2}(B³,S²).

More about the motivation, and morale:

In the setting of Yang-Mills theory in higher (than 4) dimension, one of the main questions is to prove a regularity theorem for minimizers (or stationary points) of the Yang-Mills functional in the setting where the curvature forms are compatible with extra structure on the underlying manifold (e.g. a complex structure or a G_2 structure), matching the algebraic results available in the setting of coherent reflexive sheaves.

For curvatures compatible with a complex structure, the singular set of a reflexive sheaf is known to have complex codimension 3, or real codimension 6.

The goal entails two parts, and this paper is related to part (2):

(1) understand the functional analysis implications of the nonlinearity arising from the formula F=dA+A\wedge A expressing the curvature form in terms of the connection form. Present a framework of "weak connections" over "singular spaces", in which minimization of the Yang-Mills energy can be done by the direct method of the calculus of variations. The objects appearing from this should be a weak, measure-theoretic, version of reflexive sheaves.

(2) prove the optimal regularity of weak minimizers, i.e. find how to use the information that a "weak connection" froma above is minimizing the Yang-Mills energy, in order to show that it will be a classical object except on a well-behaved singular set. Gang Tian in his paper "Gauge theory and calibrated geometry" described a framework for linking the algebraic picture to the analytic one, utilizing in n>4 dimensions a "calibrating" closed (n-4)-form \Omega to describe an analogue of the celebrated anti-self-duality, and to link the Yang-Mills equations to the structure of the singular set.

In my result above, which shows codimension-5 (and not 6 as above) singularities in dimension n=5, one such calibrating 1-form is present. However the clibrating form is dr, where r is the distance to the origin, so it is weakly closed but not continuous. Moreover no compatibility condition on the curvatures is imposed.

Therefore this points to the idea that the calibration's smoothness and/or the compatibility condition with a complex structure must be responsible for some extra regularity of Gang Tian's calibrated stationary curvatures.

In this paper (which is an improved version of a part of my PhD thesis at ETH) we introduce the functional analysis setting for the minimization of the Yang-Mills energy in higher dimension, and we prove the "partial regularity" result in 5 dimensions stating that minimizers are locally regular classical bundles in the interior of the domain except at a set of isolated singular points.

For an introduction to the subject see the lecture notes

The Variations of Yang-Mills Lagrangian, by Tristan Rivière ( direct link to his pdf file )

The space of weak connections on singular bundles and a motivation

(second order explanations in lighter grey):

The Yang-Mills energy of a connection A is the L²-energy of its curvature, ||F||². The integrand, i.e. the pointwise norm of the curvature, is invariant under gauge transformations, unlike the Sobolev norm of the connection form. This is due to the cancellations appearing in the expression of the curvature form.

(To explain these cancellations, one can think of the connection form as the covariant derivation, locally of the form d+A, i.e. a perturbation of the usual exterior derivative d. The curvature form is then encoding the result of applying this perturbed exterior derivative twice, (d+A)². The cancellations providing the gauge-invariance of the curvature can be interpreted as the remnant of the well-known formula d²=0 which persist after perturbation.)

We consider the following problem: fix the bundle and the connection over the boundary of a 5-dimensional manifold M⁵ up to the action of the set of measurable gauge changes. Then we minimize the Yang-Mills energy of possible extensions of such bundles with connection over the interior of M⁵. The results of the paper re formulated in the case of M⁵=B⁵, the unit ball in Euclidean space.

In general the minimization necessites a more general space of connections than connections which locally in some gauge have Sobolev coefficients in W^{1,2}, because we expect minimizers to have indeed topological type singularities (which a posteriori is proved by the example in my above paper A singular radial connection over B^5 minimizing the Yang-Mills energy).

By definition we have a topological singularity at a point p, if the curvature over the small 4-sphere \partial B_\epsilon(p) has nontrivial Chern class.

Here is why this contradicts the fact that in some local gauge over B_\epsilon(p) the connection form A has W^{1,2} coefficients: If such gauge by contradiction existed, we could make sense in the sense of distributions of the Chern-Simons formula for F, which formally states that the Chern-Weil 4-form tr(F\wedgeF) is exact and expressable in terms of A ...then using the fact that an exact form is zero on a cycle we would have a zero-Chern-number bundle over \partial B_\epsilon(p), a contradiction to the fact that p was a topological singularity!

The class of generalized connections which we introduce is defined grosso-modo by considering L²-connection forms and asserting that dA+A\wedge A must also be L² and "for almost all" 4-dimensional slices by closed submanifolds of B⁵, the restriction of the connection to such 4D slice is W^{1,2} in local gauges. This is reminiscent of the Ambrosio-Kirchheim result which defines rectifiable integer multiplicity metric k-currents in terms of their codimension-k slices.

Using the L²-norm of the curvature we may then bound the Hölder-1/2 oscillation of such slices done on 4-spheres with respect to the Donaldson distance between curvatures on the model S⁴, and this bound allows to pass to the limit bounded-curvature sequences of weak connections, proving the closure result, and the existence of minimizing weak curvatures.

Behind the above Hölder estimate is an approximation result for weak curvature forms in the strong L² norm by curvatures which are classical smooth objects locally outside finitely many points.

The approximation result

The approximation result is obtained by considering a good family of ball coverings of the domain into cubes of size of order \epsilon. We obtain that the weak connection form A and its distributional curvature form F are approximated strongly in L² as the scale \epsilon goes to zero.

We choose the balls so that the centers are on an epsilon-grid and slicing A, F by their boundaries we get slices for which A is W^{1,2} and A, B are bounded in L² on the boundaries, via a Fubini theorem, by the volume integrals of A, F.

If F has small rescaled energy (these balls are called ¨good balls¨) then Uhlenbeck`s epsilon-regularity gives a well-controlled Coulomb gauge on each such boundary and we proceed to do an extension of the connection`s restrictions and of the Uhlenbeck gauge from the boundary to the interior. The extension is modelled on harmonic extension, but some extra care (including an adapted version of Uhlenbeck's result) must be used to ensure good behavior of the components of A normal to the boundary.

If the above epsilon-regularity is not available, i.e. we have relatively large rescaled energy of F on the boundary, then we have "bad balls", where we introduce a singularity. By a covering argument the total volume of bad balls goes to zero, so we just need a less efficient bound for the extension on them.

The good bounds on good balls and the bounds on bad balls allow to conclude via a dominated convergence argument.

The partial regularity

For the partial regularity of minimizers and of stationary points belonging to our new weak connection space follows from previous work by Yves Meyer and Tristan Rivière, using a second, refined, version of our approximation result. Meyer and Rivière proved that stationary critical points of the Yang-Mills functional are smooth in small (renormalized) energy regions provided they are approximatable in the natural Morrey norm by smooth connections on classical bundles.

The need for smooth approximation is due to the fact that Meyer-Rivière use a Morrey version of Uhlenbeck's epsilon-regularity, which is based on the fact that if the scale-invariant energy is small then "replacing the regularity space" by arbitrarily little improvement will provide the space of connections with a Banach manifold structure.

This worked in the case of Sobolev spaces by considering W^{2,2+\epsilon} gauges and W^{1,2+\epsilon} connection forms and obtaining as \epsilon=0 the spaces where the problem is really set. In that case the Banach manifold structure on the space of gauges comes because W^{2,2+\epsilon} gauges are continuous for \epsilon>0.

Replacing Sobolev spaces by the correct scale-invariant Morrey spaces has teh same effect of allowing to pass from space of critical noncontinuous gauges to continuous ones just by perturbing the integrability exponent, however in that case the limit \epsilon\to 0 is not anymore justified, and the approximability by smooth connections had to be assumed as a hypothesis by Meyer-Rivière. The approximability assumption is also present in Tao-Tian.

Our approximation result allows to prove this approximability in case of small Morrey energy, thus removing the assumption from precedent works.

We prove here an extension of Uhlenbeck's epsilon-regularity result for curvatures on SU(2)-bundles in 4 dimensions: Uhlenbeck's result assumed that the curvature's L^2 norm were smaller than a constant \epsilon and deduced the existence of W^{2,2} Coulomb gauges for which a W^{1,2} bound on the connection form by the curvature was insured. As a result of this paper we n thihave that the epsilon-smallness assumption can be removed, but as a consequence we have bounds in a Sobolev-Lorentz norm instead of a Sobolev norm.

More precisely we have bounds for the differential of the gauge (and thus for the connection) in L^{4,\infty} instead of L^4. Since we are in dimension 4, this has the consequence of allowing the singularities of the form x/|x| which were not allowed before, but their number is quantitatively controlled, because we use the limit norm where one such singularity has norm of order 1.

The main tool for this result is the existence of Sobolev-Lorentz-controlled nonlinear extensions of functions u in W^{1,n}(S^n, S^n). This result is proved for n=1,2,3, and for n=3 we have S^3=SU(2), which allows to prove the above global gauge existence. The precise statement is that we may find extensions of such u to B^{n+1} with values in S^n whose gradient has L^{n+1,\infty} norm controlled only in terms of the L^n norm of the gradient of u. The Lorentz space estimate is optimal, as discussed in the first section.

This kind of nonlinear, controlled, Sobolev extension question in critical spaces appears in this article for the first time, so far as I know.

The difficulty is given by the nonlinearity: while a natural, harmonic type, extension would give the correct bounds, or even the better L^{n+1} bounds, the harmonic extension does not respect the constraint of having values in S^n.

If we had a way to insure that the values of some controlled extension stay at finite distance from the center of the sphere S^n, then we would be able to reproject such extension to the sphere, keeping the good norm estimates.

However in order to get this kind of control, a more detailed discussion and scale separation is required.

For n=3, the methods used mix a variety of techniques, from the conformal invariance of the energies involved, to the "good center projection" trick by Hardt-Kinderlehrer-Lin, to the product structure identifying S^3 with SU(2).

For n=2 we have a much more elegant proof based on Hopf lifts, and inspired from previoous work of Hardt-Rivière. In this case the norm bounds are of quadratic growth, which we think is optimal. Intead in the n=3 case we just have double exponential growth for the bound of the extension's norm in terms of the boundary value's norm.

We identify three reformulations of a transportation model with the penalization of congestion, in the case where the objects to be transported are not measures, but rather more general distributions belonging to the natural spaces where the problem makes sense.

Besides the primal and dual formulations of the transportation problems (called Beckman and Wardrop problems), we describe a third one based on Smirnov's decomposition of the underlying generalized vector fields. It consists of minimizing a functional on measures on the space of arcs, under a constraint on the distribution of endpoints under those measures.

  • arxiv.org/abs/1204.0209 Interior partial regularity for minimal L^p-vectorfields with integer fluxes To appear in Ann. S.N.S. Pisa Cl. Sci. DOI

Here I study the regularity for minimizing vector fields with integer fluxes in R^3.

We proved with Tristan Rivière in Weak closure of singular abelian L^p-bundles in 3-dimensions that such minimzers exist, and the integrality condition in dimension 3 can be interpreted as a "number of turns" around singular points, counted by a topological invariant, the second Chern class of a corresponding U(1)-connection.

The partial regularity result is proved here by an unusual method: instead of considering norm estimates and proving the epsilon-regularity by improving regularity estimates in function spaces, we rather reduce to a combinatorial problem, where the question can be solved by a combinaorial reasoning (based on the maxflow-mincut theorem).

Sketch of the reduction to a combinatorial problem:

To prove epsilon-regularity, we look at a minimizer on a ball having small rescaled boundary energy. We approximate such minimizer in the strong norm by a vector field which has just finitely many singular points of integer degree.

Then we apply Smirnov's decomposition theorem (from his 1994 article), representing such vector field as a superposition of 1-currents of integration along arcs, with a "no-cancellations" condition. We then partition the set of arcs into finitely many Borel sets, according to where the end points and start points of those arcs are (e.g. at a given singularity or at the boundary of the ball).

To a Borel set of arcs we associate a directed edge in a graph (over nodes labelled in correspondence to where the start/end points of arcs in that Borel set were). On such edge we attach a weight equal to the "total flux" flowing through the superposition of arcs from that set.

Then any linear combination of indicator functions of these Borel sets with coefficients of absolute value less than one will give a vector field with norm bounded by the original one. If we represent the Borel sets by edges of a finite graph, this corresponds to a kind of "capacity constraint".

We can study the problem at two levels: the level of vector fields which are superposition of some Borel sets of arcs, and the level of finite graphs with orientations and weights. We construct a smaller energy competitor at the vector field level by requring the capacity constraint at the graph level. We ask that no divergence is created at the level of vector fields by requiring the Kirchoff condition at the graph level. This means that we seek a "flow with capacity constraint" at the graph level.