arXiv subjects

Aaron Naber

Publications and source records attributed to Aaron Naber.

At least 37 records · Page 2Linked to original sources

arXiv2018

Rectifiability of Singular Sets in Noncollapsed Spaces with Ricci Curvature bounded below

This paper is concerned with the structure of Gromov-Hausdorff limit spaces $(M^n_i,g_i,p_i)\stackrel{d_{GH}}{\longrightarrow} (X^n,d,p)$ of Riemannian manifolds satisfying a uniform lower Ricci curvature bound $Rc_{M^n_i}\geq -(n-1)$ as well as the noncollapsing assumption $Vol(B_1(p_i))>v>0$. In such cases, there is a filtration of the singular set, $S_0\subset S_1\cdots S_{n-1}:= S$, where $S^k:= \{x\in X:\text{ no tangent cone at $x$ is }(k+1)\text{-symmetric}\}$; equivalently no tangent cone splits off a Euclidean factor $\mathbb{R}^{k+1}$ isometrically. Moreover, by \cite{ChCoI}, $\dim S^k\leq k$. However, little else has been understood about the structure of the singular set $S$. Our first result for such limit spaces $X^n$ states that $S^k$ is $k$-rectifiable. In fact, we will show that for $k$-a.e. $x\in S^k$, {\it every} tangent cone $X_x$ at $x$ is $k$-symmetric i.e. that $X_x= \mathbb{R}^k\times C(Y)$ where $C(Y)$ might depend on the particular $X_x$. We use this to show that there exists $ε=ε(n,v)$, and a $(n-2)$-rectifible set $S^{n-2}_ε$, with finite $(n-2)$-dimensional Hausdorff measure $H^{n-2}(S_ε^{n-2})<C(n,v)$, such that $X^n\setminus S^{n-2}_ε$ is bi-Hölder equivalent to a smooth riemannian manifold. This improves the regularity results of \cite{ChCoI}. Additionally, we will see that tangent cones are unique of a subset of Hausdorff $(n-2)$ dimensional measure zero. Our analysis is based on several new ideas, including a sharp cone-splitting theorem and a geometric transformation theorem, which will allow us to control the degeneration of harmonic functions on these neck regions.

math.DG↗

arXiv2017

The Singular Structure and Regularity of Stationary and Minimizing Varifolds

If one considers an integral varifold $I^m\subseteq M$ with bounded mean curvature, and if $S^k(I)\equiv\{x\in M: \text{ no tangent cone at $x$ is }k+1\text{-symmetric}\}$ is the standard stratification of the singular set, then it is well known that $\dim S^k\leq k$. In complete generality nothing else is known about the singular sets $S^k(I)$. In this paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum $S^k(I)$ is $k$-rectifiable. In fact, we prove for $k$-a.e. point $x\in S^k$ that there exists a unique $k$-plane $V^k$ such that every tangent cone at $x$ is of the form $V\times C$ for some cone $C$. In the case of minimizing hypersurfaces $I^{n-1}\subseteq M^n$ we can go further. Indeed, we can show that the singular set $S(I)$, which is known to satisfy $\dim S(I)\leq n-8$, is in fact $n-8$ rectifiable with uniformly finite $n-8$ measure. An effective version of this allows us to prove that the second fundamental form $A$ has apriori estimates in $L^7_{weak}$ on $I$, an estimate which is sharp as $|A|$ is not in $L^7$ for the Simons cone. In fact, we prove the much stronger estimate that the regularity scale $r_I$ has $L^7_{weak}$-estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative stratifications $S^k_{ε,r}$ and $S^k_ε\equiv S^k_{ε,0}$. Roughly, $x\in S^k_ε\subseteq I$ if no ball $B_r(x)$ is $ε$-close to being $k+1$-symmetric. We show that $S^k_ε$ is $k$-rectifiable and satisfies the Minkowski estimate $Vol(B_r\,S_ε^k)\leq C_εr^{n-k}$. The proof requires a new $L^2$-subspace approximation theorem for integral varifolds with bounded mean curvature, and a $W^{1,p}$-Reifenberg type theorem proved by the authors in \cite{NaVa+}.

math.DG↗

arXiv2017

Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds

We prove that a metric measure space $(X,d,m)$ satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space $W^{1,2}$ is Hilbert is rectifiable. That is, a $RCD^*(K,N)$-space is rectifiable, and in particular for $m$-a.e. point the tangent cone is unique and euclidean of dimension at most $N$. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. To this aim we also show a sharp integral Abresh-Gromoll type inequality on the excess function and an Abresh-Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting.

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arXiv2016

Stratification for the singular set of approximate harmonic maps

The aim of this note is to extend the results in arXiv:1504.02043 to the case of approximate harmonic maps. More precisely, we will proved that the singular strata $S^k(u)$ of an approximate harmonic map are k-rectifiable, and we will show effect bounds on the quantitative strata. In the process we will simplify many of the arguments from arXiv:1504.02043, and in particular we produce a new main covering lemmas which vastly simplifies the older argument.

math.DG↗

arXiv2016

Quantitative regularity for p-harmonic maps

In this article, we study the regularity of minimizing and stationary $p$-harmonic maps between Riemannian manifolds. The aim is obtaining Minkowski-type volume estimates on the singular set $S(f)=\{x \ \ s.t. \ \ f \text{ is not continuous at } x\}$, as opposed to the weaker and non quantitative Hausdorff dimension bounds currently available in literature for generic $p$. The main technique used in this paper is the quantitative stratification, which is based on the study of the approximate symmetries of the tangent maps of $f$. In this article, we generalize the study carried out in \cite{ChNa2} for minimizing $2$-harmonic maps to generic $p\in (1,\infty)$. Moreover, we analyze also the stationary case where the lack of compactness makes the study more complicated. In order to understand the degeneracy intrinsic in the behaviour of stationary maps, we study the defect measure naturally associated to a sequence of such maps and generalize the results obtained in \cite{lin_stat}. By using refined covering arguments, we also improve the estimates in the case of isolated singularities and obtain a definite bound on the number of singular points. This result seems to be new even for minimizing $2$-harmonic maps.

math.AP↗

arXiv2016

Volume estimates on the critical sets of solutions to elliptic PDEs

In this paper we study solutions to elliptic linear equations $L(u)=\partial_i(a^{ij}(x)\partial_j u) + b^i(x) \partial_i u + c(x) u=0$, either on $R^n$ or a Riemannian manifold, under the assumption of Lipschitz control on the coefficients $a^{ij}$. We focus our attention on the critical set $Cr(u)\equiv\{x:|\nabla u|=0\}$ and the singular set $S(u)\equiv\{x:u=|\nabla u|=0\}$, and more importantly on effective versions of these. Currently, under the coefficient control we have assumed, the strongest results in the literature say that the singular set is n-2-dimensional, however at this point it has not even been shown that $H^{n-2}(S)<\infty$ unless the coefficients are smooth. Fundamentally, this is due to the need of an $ε$-regularity theorem which requires higher coefficient control as the frequency increases. We introduce new techniques for estimating the critical and singular set, which avoids the need of any such $ε$-regularity. Consequently, we prove that if the frequency of u is bounded by $Λ$ then we have the estimates $H^{n-2}(C(u))\leq C^{Λ^2}$, $H^{n-2}(S(u))\leq C^{Λ^2}$, depending on whether the equation is critical or not. More importantly, we prove corresponding estimates for the {\it effective} critical and singular sets. Even under the assumption of analytic coefficients these results are much sharper than those currently in the literature. We also give applications of the technique to the nodal set of solutions, and to give estimates on the corresponding eigenvalue problem.

math.AP↗

arXiv2016

Ricci Curvature and Bochner Formulas for Martingales

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical Bochner formula on M is related to lower bounds on Ricci curvature. Using this formalism, we obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and streamlined proofs of the previous characterizations of bounded Ricci curvature of the second author (arXiv:1306.6512).

math.DG↗

arXiv2016

Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps

In this paper we study the regularity of stationary and minimizing harmonic maps $f:B_2(p)\subseteq M\to N$ between Riemannian manifolds. If $S^k(f)\equiv\{x\in M: \text{ no tangent map at $x$ is }k+1\text{-symmetric}\}$ is $k^{th}$-stratum of the singular set of $f$, then it is well known that $\dim S^k\leq k$, however little else about the structure of $S^k(f)$ is understood in any generality. Our first result is for a general stationary harmonic map, where we prove that $S^k(f)$ is $k$-rectifiable. In the case of minimizing harmonic maps we go further, and prove that the singular set $S(f)$, which is well known to satisfy $\dim S(f)\leq n-3$, is in fact $n-3$-rectifiable with uniformly {\it finite} $n-3$-measure. An effective version of this allows us to prove that $|\nabla f|$ has estimates in $L^3_{weak}$, an estimate which is sharp as $|\nabla f|$ may not live in $L^3$. The above results are in fact just applications of a new class of estimates we prove on the {\it quantitative} stratifications $S^k_{ε,r}(f)$ and $S^k_ε(f)\equiv S^k_{ε,0}(f)$. Roughly, $S^k_ε\subseteq M$ is the collection of points $x\in S^k_ε$ for which no ball $B_r(x)$ is $ε$-close to being $k+1$-symmetric. We show that $S^k_ε$ is $k$-rectifiable and satisfies the Minkowski estimate $Vol(B_r\,S_ε^k)\leq C r^{n-k}$. The proofs require a new $L^2$-subspace approximation theorem for stationary harmonic maps, as well as new $W^{1,p}$-Reifenberg and rectifiable-Reifenberg type theorems. These results are generalizations of the classical Reifenberg, and give checkable criteria to determine when a set is $k$-rectifiable with uniform measure estimates. The new Reifenberg type theorems may be of some independent interest.

math.DG↗

arXiv2015

Regularity of Einstein Manifolds and the Codimension 4 Conjecture

In this paper, we are concerned with the regularity of noncollapsed Riemannian manifolds $(M^n,g)$ with bounded Ricci curvature, as well as their Gromov-Hausdorff limit spaces $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow} (X,d)$, where $d_j$ denotes the Riemannian distance. Our main result is a solution to the codimension $4$ conjecture, namely that $X$ is smooth away from a closed subset of codimension $4$. We combine this result with the ideas of quantitative stratification to prove a priori $L^q$ estimates on the full curvature $|Rm|$ for all $q<2$. In the case of Einstein manifolds, we improve this to estimates on the regularity scale. We apply this to prove a conjecture of Anderson that the collection of $4$-manifolds $(M^4,g)$ with $|Ric_{M^4}|\leq 3$, $Vol(M)>v>0$, and $diam(M)\leq D$ contains at most a finite number of diffeomorphism classes. A local version of this is used to show that noncollapsed $4$-manifolds with bounded Ricci curvature have a priori $L^2$ Riemannian curvature estimates.

math.DG↗

arXiv2015

Weak solutions for the Ricci flow I

This is the first of a series of papers, where we introduce a new class of estimates for the Ricci flow, and use them both to characterize solutions of the Ricci flow and to provide a notion of weak solutions to the Ricci flow in the nonsmooth setting. In this first paper, we prove various new estimates for the Ricci flow, and show that they in fact characterize solutions of the Ricci flow. Namely, given a family $(M,g_t)_{t\in I}$ of Riemannian manifolds, we consider the path space $P\mathcal{M}$ of its space time $\mathcal{M}=M\times I$. Our first characterization says that $(M,g_t)_{t\in I}$ evolves by Ricci flow if and only if an infinite dimensional gradient estimate holds for all functions on $P\mathcal{M}$. We prove additional characterizations in terms of the $C^{1/2}$-regularity of martingales on path space, as well as characterizations in terms of log-Sobolev and spectral gap inequalities for a family of Ornstein-Uhlenbeck type operators. Our estimates are infinite dimensional generalizations of much more elementary estimates for the linear heat equation on $(M,g_t)_{t\in I}$, which themselves generalize the Bakry-Emery-Ledoux estimates for spaces with lower Ricci curvature bounds. Based on our characterizations we can define a notion of weak solutions for the Ricci flow. We will develop the structure theory of these weak solutions in subsequent papers.

math.DG↗

arXiv2015

Characterizations of Bounded Ricci Curvature on Smooth and NonSmooth Spaces

There are two primary goals to this paper. In the first part of the paper we study smooth metric measure spaces (M^n,g,e^{-f}dv_g) and give several ways of characterizing bounds -Kg\leq \Ric+\nabla^2f\leq Kg on the Ricci curvature of the manifold. In particular, we see how bounded Ricci curvature on M controls the analysis of path space P(M) in a manner analogous to how lower Ricci curvature controls the analysis on M. In the second part of the paper we develop the analytic tools needed to in order to use these new characterizations to give a definition of bounded Ricci curvature on general metric measure spaces (X,d,m). We show that on such spaces many of the properties of smooth spaces with bounded Ricci curvature continue to hold on metric-measure spaces with bounded Ricci curvature.

math.DG↗

arXiv2015

Topology and ε-regularity Theorems on Collapsed Manifolds with Ricci Curvature Bounds

In this paper we discuss and prove $ε$-regularity theorems for Einstein manifolds $(M^n,g)$, and more generally manifolds with just bounded Ricci curvature, in the collapsed setting. A key tool in the regularity theory of noncollapsed Einstein manifolds is the following: If $x\in M^n$ is such that $Vol(B_1(x))>v>0$ and that $B_2(x)$ is sufficiently Gromov-Hausdorff close to a cone space $B_2(0^{n-\ell},y^*)\subset \mathbb{R}^{n-\ell}\times C(Y^{\ell-1})$ for $\ell\leq 3$, then in fact $|Rm|\leq 1$ on $B_1(x)$. No such results are known in the collapsed setting, and in fact it is easy to see without more such results are false. It turns out that the failure of such an estimate is related to topology. Our main theorem is that for the above setting in the collapsed context, either the curvature is bounded, or there are topological constraints on $B_1(x)$. More precisely, using established techniques one can see there exists $ε(n)$ such that if $(M^n,g)$ is an Einstein manifold and $B_2(x)$ is $ε$-Gromov-Hausdorff close to ball in $B_2(0^{k-\ell},z^*)\subset\mathbb{R}^{k-\ell}\times Z^\ell$, then the fibered fundamental group $Γ_ε(x)\equiv Image[π_1(B_ε(x))\toπ_1(B_2(x))]$ is almost nilpotent with $rank(Γ_ε(x))\leq n-k$. The main result of the this paper states that if $rank(Γ_ε(x))= n-k$ is maximal, then $|Rm|\leq C$ on $B_1(x)$. In the case when the ball is close to Euclidean, this is both a necessary and sufficient condition. There are generalizations of this result to bounded Ricci curvature and even just lower Ricci curvature.

math.DG↗

arXiv2014

Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the p-Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $Δ_p$ when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces.

math.DG↗

arXiv2013

Quantitative Stratification and the Regularity of Harmonic Map Flow

In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider H^1_loc-maps u defined on a parabolic ball P\subset M\times R and with target manifold N, that have bounded Dirichlet-energy and Struwe-energy. We define a quantitative stratification, which groups together points in the domain into quantitative weakly singular strata S^j_{η,r}(u) according to the number of approximate symmetries of u at certain scales, and prove that their tubular neighborhoods have small volume, namely Vol(T_r(\cS^j_{η,r}(u))< Cr^{m+2-j-\eps}. In particular, this generalizes the known Hausdorff estimate dim S^j(u)< j for the weakly singular strata of suitable weak solutions of the harmonic map flow. As an application, specializing to Chen-Struwe solutions with target manifolds that do not admit certain harmonic and quasi-harmonic spheres, we obtain refined Minkowski estimates for the singular set. This generalizes a result of Lin-Wang. We also obtain L^p-estimates for the reciprocal of the regularity scale. The results are analogous to our results for mean curvature flow that we recently proved.

math.DG↗

arXiv2013

Critical sets of elliptic equations

Given a solution $u$ to a linear homogeneous second order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set $\Cr(u)\equiv \{x:|\nabla u|(x)=0\}$. The results are new even for harmonic functions on $\dR^n$. Given such a $u$, the standard {\it first order} stratification $\{\cS^k\}$ of $u$ separates points $x$ based on the degrees of symmetry of the leading order polynomial of $u-u(x)$. In this paper we give a quantitative stratification $\{\cS^k_{η,r}\}$ of $u$, which separates points based on the number of {\it almost} symmetries of {\it approximate} leading order polynomials of $u$ at various scales. We prove effective estimates on the volume of the tubular neighborhood of each $\cS^k_{η,r}$, which lead directly to $(n-2+ε)$-Minkowski content estimates for the critical set of $u$. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to a uniform $(n-2)$-Hausdorff measure estimate on the critical set of $u$.

math.DG↗

arXiv2012

Lower Bounds on Ricci Curvature and Quantitative Behavior of Singular Sets

Let Y^n denote the Gromov-Hausdorff limit of a sequence M^n_i-> Y^n of v-noncollapsed riemannian manifolds with Ric_i\geq-(n-1). The singular set S of Y has a stratification S^0\subset S^1\subset\...\subset S, where y\in S^k if no tangent cone at y splits off a factor R^{k+1} isometrically. There is a known Hausdorff dimension bound dimS^k\leq k. Here, we define for all η>0, 0<r\leq 1, the {\it k-th effective singular stratum} S^k_{η,r} such that \bigcup_η\bigcap_r \,\cS^k_{η,r}= \cS^k. Sharpening the bound dim S^k\leq k, we prove that the r-tubular neighborhood satisfies: Vol(T_r(S^k_{η,r})\cap B_{1/2}(y))\leq c(n,v,η)r^{n-k-η}, for all y. The proof depends on a {\it quantitative differentiation} argument; for further explanation, see Section 2. The applications give new curvature estimates for Einstein manifolds. Let Rm denote the curvature tensor and regard |Rm(y)|= \infty unless Y^n is smooth in some neighborhood of y. Put \cB_r=\{y: |sup_{B_r(y)}|Rm|\geq r^{-2}\}. Assuming in addition that the M^n_i are Kähler-Einstein with ||Rm||_{L_2}\leq C, we get the volume bound Vol(\cB_r\cap B_{1/2}(y))\leq c(n,v,C)r^4$ for all y. In the Kähler-Einstein case, without assuming any integral curvature bound on the M^n_i, we obtain a slightly weaker volume bound on \cB_r, which yields an a priori L_p curvature bound for all p<2; see Section 1 the for the precise statement (which is sharper).

math.DG↗

arXiv2012

Quantitative Stratification and the Regularity of Harmonic Maps and Minimal Currents

We introduce techniques for turning estimates on the infinitesimal behavior of solutions to nonlinear equations (statements concerning tangent cones and blow ups) into more effective control. In the present paper, we focus on proving regularity theorems for stationary and minimizing harmonic maps and minimal currents. There are several aspects to our improvements of known estimates. First, we replace known estimates on the Hausdorff dimension of singular sets by estimates on their Minkowski r-content, or equivalently, on the volumes of their r-tubular neighborhoods. Second, we give improved regularity control with respect to the number of derivatives bounded and/or on the norm in which the derivatives are bounded. As an example of the former, our results for minimizing harmonic maps f : M->N between riemannian manifolds include a priori bounds in W^{1,p}\cap W^{2,p} for all p < 3. These are the first such bounds involving second derivatives in general dimensions. Finally, the quantity we control is actually provides much stronger information than follows from a bound on the L^p norm of derivatives. Namely, we obtain L^p bounds for the inverse of the regularity scale r_f(x)=max{r:sup_{B_r(x)}(r|\nabla f|+r^2|\nabla^2 f|)\leq 1}. Applications to minimal hypersufaces include a priori L^p bounds for the second fundamental form A for all p < 7. Previously known bounds were for p < 2+ε(n). Again, the full theorem is much stronger and yields Lp bounds for the corresponding regularity scale r_A(x)=max{r:sup_{B_r(x)}rA\leq 1}. In outline, our discussion follows that of an earlier paper in which we proved analogous estimates in the context of noncollapsed riemannian manifolds with a lower bound on Ricci curvature. These were applied to Einstein manifolds. A key point in all of these arguments is to establish the relevant quantitative differentiation theorem.

math.DG↗

arXiv2012

Quantitative Stratification and the Regularity of Mean Curvature Flow

Let $\cM$ be a Brakke flow of $n$-dimensional surfaces in $R^N$. The singular set $\cS\subset\cM$ has a stratification $\cS^0\subset\cS^1\subset...\cS$, where $X\in \cS^j$ if no tangent flow at $X$ has more than $j$ symmetries. Here, we define quantitative singular strata $\cS^j_{η,r}$ satisfying $\cup_{η>0}\cap_{0<r} \cS^j_{η,r}=\cS^j$. Sharpening the known parabolic Hausdorff dimension bound $\dim \cS^j\leq j$, we prove the effective Minkowski estimates that the volume of $r$-tubular neighborhoods of $\cS^j_{η,r}$ satisfies $\Vol (T_r(\cS^j_{η,r})\cap B_1)\leq Cr^{N+2-j-\varepsilon}$. Our primary application of this is to higher regularity of Brakke flows starting at $k$-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of $k$-convex hypersurfaces, any backwards selfsimilar limit flow with at least $k$ symmetries is in fact a static multiplicity one plane. Then, denoting by $\cB_r\subset\cM$ the set of points with regularity scale less than $r$, we prove that $\Vol(T_r(\cB_r))\leq C r^{n+4-k-\varepsilon}$. This gives $L^p$-estimates for the second fundamental form for any $p<n+1-k$. In fact, the estimates are much stronger and give $L^p$-estimates for the inverse of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (arXiv:1103.1819v3) and Cheeger and Naber (arXiv:1107.3097v1).

math.DG↗