Northwestern University, U.S.A.

**Office:**Lunt 220- email: anaber(at)math.northwestern.edu

Areas of current research:

My current research interests focus on the study of geometrically motivated equations and their applications. This includes work in the areas of Ricci curvature, nonlinear harmonic maps, yang-mills, minimal varifolds, sectional curvature, ricci solitons, mean curvature flow, Ricci flow, and general elliptic equations. A rough outline of some of my research is below, see CV for a more complete and updated list.

**Yang-Mills and Energy Quantization:**

**Energy Identity and L^1 Hessian Conjectures:**If A_i are a sequence of stationary Yang-Mills connections, then one can pass to a weak limit A_i->A. In such a limiting process one inevitably must deal with blow up, and a wonderful way to understand this is through the defect measure. That is, if F_i are the curvatures of A_i, then one can show that the measures |F_i|^2 dv_i -> |F_A|^2dv + e(x)dH^{n-4}_S, where S is an n-4 rectifiable set and H^{n-4}_S is the hausdorff measure on S. In short, the sequence A_i may only blow up along a well behaved n-4 dimensional subset, and when this happens the energy concentrates along this subset. The energy density e(x) measures how much energy was lost in this blow up, and it has been a conjectural picture (which has been proved in many special cases) that e(x) may be computed explicitly as the sum of bubble energies arising from the sequence at x. In [NV16] together with Daniele Valtorta we prove this to hold in generality. Additionally, we show that for a general stationary connection A there is an apriori L^1 hessian bound \int |\nabla^2 F_A|^2 < C on the curvature. These two points are actually proved simultaneously. The proof involves a new quantitative bubble tree decomposition, which decomposes a general fixed solution into so called quantitative annular and bubble regions in an effective manner. The most difficult analysis is on the annular regions, and dealing with them involves introducing a new type of gauge condition, which we call an eps-gauge. This eps-gauge generalizes the coulomb gauge, but exists even over singular regions, and allows us to do analysis over the singularities.

**Stratification and Regularity Theory:**

**Rectifiability of Singular Sets of Nonlinear Harmonic Maps**: If f:M->N is a stationary harmonic map, then one can define the stratification S^k(f) = {x: no tangent map at x is k+1 symmetric}. It is a classical result of Schoen/Uhlenbeck that dimS^k <= k, however essentially nothing else has been understood in generality. If f is also assumed to be minimizing and N is an analytic manifold, then it has been proven by Simon that S^{n-3} is rectifiable. Together with Daniele Valtorta, in [NV15] we prove for a general stationary harmonic map into any target that all the S^k are k-rectifiable, which is roughly to say that S^k are k-manifolds away from a set of measure zero. The techniques of this paper include the introduction of a new W^{1,p}-Reifenberg and rectifiable-Reifenberg theorem, which are of some independent interest. Roughly, these give criteria which ensure that a set is rectifiable with measure estimates. In order to apply these theorems to the singular set, we prove a new L^2-subspace approximation theorem for stationary harmonic maps. It is important in the argument that we do not deal directly with the stratification, but instead provide effective estimates for the quantitative stratifications, a notion first introduced in [ChN]. It is worth pointing out that the techniques are very general, and as a vague rule tend to work whenever a Federer dimension reduction argument works.**n-3 Finiteness and Sharp Schauder Estimates for Nonlinear Harmonic Maps:**If f is further assumed to be a minimizing harmonic map, then in [NV15] we get effective estimates on the n-3 Hausdorff measure of the singular set. More analytically, we prove |\nabla f| has effective estimates in L^3_{weak}, which is sharp since there are examples which do not live in L^3. The results are related to the rectifiability methods above, as we in fact prove for even stationary harmonic maps very effective estimates on the quantitative stratifications. These estimates hold for any stationary map, but in the case of minimizers one can show for the top stratum that the quantitative stratification and the classical stratification agree, and hence one obtains effective estimates on the singular set itself.

**Structure of Singular Sets of Varifolds with Bounded Mean Curvature:**As another example of the above technique, if I^m is an integral varifold with bounded mean curvature, then one can define the stratification S^k(I) = {x: no tangent cone at x is k+1 symmetric}. It was proven by Federer that dimS^k<=k, where the dimension is in the Hausdorff sense. If I^m is further assumed to be a Z_2-minimizing varifold, then it was proven by Simon that S^{m-2} is rectifiable. However, besides these results nothing else about the structure of the singular set is known. In [NV15_2] we prove for an integral varifold with bounded mean curvature, in any manifold, that S^k is k-rectifiable for all k. More than that, we see that for k a.e. point x\in S^k that there exists a unique k-plane V such that 'every' tangent cone at x is of the form VxC for some cone C. That is, a.e. point of S^k has maximal symmetries, and the k-plane of symmetry is well defined independent of the tangent cone. If I is further assumed to be a minimizing hypersurface, we prove that Sing(I) is rectifiable with effective m-7 Hausdroff measure bounds. More analytically, we show that the second fundamental form |A| has apriori estimates in weak L^7. This is sharp, as the Simons and Lawson cones have the property that |A| is not even locally in L^7. As in the harmonic maps case, the techniques of this paper include a new W^{1,p}-Reifenberg and rectifiable-Reifenberg theorem, which are of some independent interest. Roughly, these give criteria which ensure that a set is rectifiable with measure estimates. In order to apply these theorems to the singular set of an integral varifold, we prove new L^2-subspace approximation theorems for integral varifolds with bounded mean curvature.**Other Applications of Quantitative Stratification and Rectifiability Methods:**The quantitative stratification was introduced in [ChN] in the context of Ricci bounds in order to give some effective estimates on singular sets and solutions. The techniques of [ChN] have since been extended to harmonic maps and minimal surfaces [ChN2], mean curvature flows [ChHaNa], harmonic map flows [ChHaNa2], critical sets of elliptic equations [ChNaVa], biharmonic maps [BL], Q-valued harmonic maps [FMS], etc... The techniques of were vastly extended in in order to prove the rectifiable and finiteness estimates. These techniques have also recently been extended to the case of stationary Yang Mills in [W].

**Regularity and Bounded Ricci Curvature:**

**Codimension Four Conjecture:**Together with Jeff Cheeger, in [ChN2] we proved the codimension four conjecture. Roughly, we show that a metric space X which is a Gromov-Hausdorff limit of noncollapsed manifolds with bounded Ricci curvature must be smooth away from a set of codimension four. Combining this with the ideas of quantitative stratification we prove L^p estimates for manifolds with bounded Ricci curvature for all p<2. One of the primary innovations toward the proof is a new transformation estimate. In short, what makes the problem difficult are in fact the codimension two singularities. Since this is a set of capacity zero, standard estimates for harmonic functions have a hard time seeing them. To solve this, it is necessary to estimate our harmonic functions modulo transformation of the image. This allows us to see into a set whose size is just big enough to rule out the existence of codimension two singularities. Ruling out codimension three singularities is comparatively simple.

**Dimension Four and Finite Diffeomorphism Conjecture**: In [ChN2] we are also able to improve the results in dimension four. We show the collection of four manifolds with bounded Ricci, diameter, and volume have at most a finite number of diffeomorphism classes, solving a conjecture of Anderson. A local version of this proves L^2 bounds for four manifolds with bounded Ricci curvature. The proof is in that order, in that we prove the L^2 curvature bound by first showing the diffeomorphism finiteness, and then combine this with some local estimates and a Chern-Gauss-Bonnet formula to conclude the curvature bound.**L^2 Conjecture and n-4 Finiteness Conjecture:**Together with Wenshuai Jiang, in [JW] we prove that a noncollapsed manifold with bounded Ricci curvature must have an apriori L^2 bound on its curvature tensor. As a preliminary result we show that the singular set of a limit of such spaces has finite n-4 Hausdorff measure, which is a strengthening of the codimension four results and proves a conjecture of Cheeger-Colding. The proof involves several new techniques, including a neck decomposition theorem which decomposes the manifold into so called neck regions and \eps-regularity regions. The power of the decomposition are the effective n-4 content estimates on the number of pieces, which is crucial to L^2 question and the H^{n-4} finiteness of the singular set. The L^2 estimate then follows from being able to prove the corresponding estimate on each neck region, which itself requires a new superconvexity estimate, which gives a growth and decay estimate on the harmonic functions on neck regions.**eps-Regularity in the Collapsed Case:**Together with Ruobing Zhang, in [NZ] we prove the first eps-regularity theorems for spaces with collapsed spaces with bounded Ricci curvature based on Gromov-Hausdorff behavior. In the noncollapsed case it is known that if a ball is GH close to a ball in R^n, then that ball must be smooth close to a ball in R^n. In the collapsed case where the ball is close to some R^k for k<n, or even R^kxZ for some metric space Z, this need not be true. However, we prove that either the ball is smooth, or there is a topological obstruction in the fundamental group of that ball, giving the dichotomy that either an eps-regularity holds or there is a topological obstruction.

**Structure and Regularity of Lower Ricci Curvature:****Constant Dimension and Isometry Group=Lie Group Conjectures:**- Examples with Lower Ricci Curvature: In a different direction it is also important to understand to what extent examples exist which are as degenerate as possible. Unlike the Alexandrov case, e.g. limits with lower sectional curvature, it is not the case that tangent cones even need to be unique anymore, though it is always an open question as to what extent this nonuniqueness can be pushed. In [CN2] we give a characterization for the families of tangent cones which may appear at a point in a noncollapsed limit. We have two primary applications of this. First we construct limit spaces whose tangent cones at a point have singular sets of varying dimensions. In particular, this rules out the possibility of stratifying limit spaces based on tangent cones. Secondly we construct examples where differing tangent cones at a point may not even be homeomorphic. In [CN3] we provide further examples of degenerate limit spaces, as well as prove a Lipschitz structure theorem for Reifenberg spaces.

**Characterizing Bounded Ricci Curvature and Path Space:****Smooth Spaces with Bounded Ricci Curvature:**In [N13] new estimates are proved for spaces with bounded Ricci curvature, and these estimates are proved to be equivalent to bounded Ricci curvature. The first such method is an infinite dimensional generalization of the Bakry-Emery gradient estimates on path space P(M) of the manifold M. The second method studies the 1-parameter decomposition of functions on path space P(M) determined by martingales, and in particular shows that bounded Ricci curvature is equivalent to certain C^{1/2}-Holder estimates. The third method studies a family of Ornstein-Uhlenbeck operators on path space, a form of infinite dimensional laplacian, and proves that bounds on the Ricci curvature are equivalent to a spectral gap on these operators.**NonSmooth Spaces with Bounded Ricci Curvature:**Using the above as motivation, in [N13] we define the notion of bounded Ricci curvature for a general metric-measure space X. We prove a variety of structure theorems about such spaces, in particular that one can still do analysis on the path space P(X) of such spaces. We show such spaces have a lower Ricci curvature bound in the sense of Lott-Villani-Sturm.**Bochner Formula for Martingales:**In [NH16] Bob Haslhofer and I construct a new type of Bochner formula for martingales on the path space of a manifold. This Bochner formula turns out to be related to two-sided bounds on Ricci curvature in much the same way the standard Bochner formula is related to lower bounds on Ricci curvature. The method not only gives rise to (much) simpler and more natural proofs of the estimates of [N13], but may be used to provide a variety of strengthened estimates, including new L^2 hessian estimates on martingales, which seems to us to be new even in R^n.

**Characterizations of Ricci Flow**: In [HN15] we prove new estimates on Ricci flows. These estimates generalize the ideas of [N13] from the elliptic context of Einstein manifolds to the parabolic context of the Ricci flow, by studying estimates on the spacetime path space of a Ricci flow. The estimates are not only new for a Ricci flow, but it is shown that if a one parameter family of Riemannian manifolds satisfy these estimates, then the family must solve the Ricci flow. Thus, these estimates are sharp enough to characterize the Ricci flow.

**Critical Sets of Elliptic Equations:**Joint with Daniele Valtorta we study critical sets of elliptic equations on manifolds. If the coefficients are smooth it has been standard that the critical set has apriori n-2 Hausdorff measure estimates, however even under only C^k bounds on the coefficients with k<\infty this has been an open question. In [NaVa] we develop new techniques for estimating such sets, and prove the desired n-2 Hausdroff measure estimates on the critical sets of elliptic equations under only Lipschitz coefficients. The regularity assumption on the coefficients is sharp. Indeed, our estimates are much stronger and actually prove estimates for the volumes of tubes around the effective critical set.

- Ricci Solitons and Ricci Flow: Ricci solitons can be viewed as a generalization of Einstein manifolds. My primary results in this area are for shrinking solitons (the ricci soliton equivalent of having positive einstein constant). In [N] a classification is given for four dimensional shrinking solitons with bounded nonnegative curvature. In this process it is shown that shrinking solitons with bounded curvature are apriori noncollapsed and gradient. For the Ricci flow, together with Hans Hein, we proved in [HN] universal log-Sobolev estimates for the conjugate heat kernel along the Ricci flow. We used this to prove new epsilon-regularity results for the Ricci flow.
- Sectional Curvature: Joint with Gang Tian we prove two collections of results in [NaT1], [NaT2]. First we analyze the orbifolds behavior of limits of n-manifolds with bounded sectional curvature, and prove that such limit spaces are Riemannian orbifolds away from a set of Hausdorff dimension n-5. In particular, limits of four manifolds with bounded sectional curvature are always Riemannian orbifolds. We apply this to study limits which only have bounded Ricci curvature. Secondly, we construct a new structure on limit spaces with bounded sectional, which we call an N^*-bundle. This structure, as opposed to the N-structure, lives on the limit space itself and can in many ways be viewed as a dual structure whose purpose is primarily for doing analysis. As applications we prove generalizations of finite diffeomorphism theorems due to Anderson, as well as a stability theorem for pinched Ricci curvature. Namely, it turns out that for a collapsing sequence with bounded diameter and sectional curvature if the Ricci curvature is tending to zero, then the limit space is a Ricci flat orbifold. Notice that in general limits with bounded sectional can have much worse than orbifold behavior, and conversely Ricci curvature bounds not be preserved when bubbling occurs. The proof relies on a maximum principle with the N^*-bundle. The maximum principle nature of the proof is quite necessary, as if even the diameter assumption is dropped then the result is not correct even if you assume the full sectional curvature is tending to zero.

Slides for Presentations:

Last updated: May 2015 by Aaron Naber.