arXiv subjects

Aaron Naber

Publications and source records attributed to Aaron Naber.

44 records · Page 3Linked to original sources

arXiv2012

New logarithmic Sobolev inequalities and an ε-regularity theorem for the Ricci flow

In this note we prove a new ε-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \in [-T,0], the pointed entropy, with W_{x}(s) <= 0, and W_{x}(s) = 0 iff (M,g(t)) is isometric to the trivial flow on R^n. Our main theorem asserts the following: There exists an ε>0, depending only on T and on lower scalar curvature and μ-entropy bounds for (M,g(-T)), such that W_{x_0}(s) > -εimplies |Rm|< r^{-2} on P_{εr}(x,0), where r^2 = |s| and P_r(x,t) \equiv B_r(x,t)\times (t-r^2,t] is the parabolic ball. The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s-average of W_x(s). To accomplish this, we require a new log-Sobolev inequality. It is well known by Perelman that the metric measure spaces (M,g(t),dv_{g(t)}) satisfy a log-Sobolev; however we prove that this is also true for the conjugate heat kernel weighted spaces (M,g(t),H_{x}(-,t)\,dv_{g(t)}). Our log-Sobolev constants for these weighted spaces are in fact universal and sharp. The weighted log-Sobolev has other consequences as well, including an average Gaussian upper bound on the conjugate heat kernel that only depends on a two-sided scalar curvature bound.

math.DG↗

arXiv2012

Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications

Consider a limit space $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α^n$ have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of $Y$ at a point $p\in Y$ are known to be metric cones $C(X)$, however they need not be unique. Let $\barΩ_{Y,p}\subseteq\cM_{GH}$ be the closed subset of compact metric spaces $X$ which arise as cross sections for the tangents cones of $Y$ at $p$. In this paper we study the properties of $\barΩ_{Y,p}$. In particular, we give necessary and sufficient conditions for an open smooth family $Ω\equiv (X_s,g_s)$ of closed manifolds to satisfy $\barΩ=\barΩ_{Y,p}$ for {\it some} limit $Y$ and point $p\in Y$ as above, where $\barΩ$ is the closure of $Ω$ in the set of metric spaces equipped with the Gromov-Hausdorff topology. We use this characterization to construct examples which exhibit fundamentally new behaviors. The first application is to construct limit spaces $(Y^n,d_Y,p)$ with $n\geq 3$ such that at $p$ there exists for every $0\leq k\leq n-2$ a tangent cone at $p$ of the form $\RR^{k}\times C(X^{n-k-1})$, where $X^{n-k-1}$ is a smooth manifold not isometric to the standard sphere. In particular, this is the first example which shows that a stratification of a limit space $Y$ based on the Euclidean behavior of tangent cones is not possible or even well defined. It is also the first example of a three dimensional limit space with nonunique tangent cones. The second application is to construct a limit space $(Y^5,d_Y,p)$, such that at $p$ the tangent cones are not only not unique, but not homeomorphic. Specifically, some tangent cones are homeomorphic to cones over $\CC P^2\sharp\bar{\CC P}^2$ while others are homeomorphic to cones over $\Sn^4$.

math.DG↗

arXiv2011

Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces

We study here limit spaces $(M_α,g_α,p_α)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_α$ have a lower Ricci curvature bound and are volume noncollapsed. Such limits $Y$ may be quite singular, however it is known that there is a subset of full measure $\cR(Y)\subseteq Y$, called {\it regular} points, along with coverings by the almost regular points $\cap_ε\cup_r\cR_{ε,r}(Y)=\cR(Y)$ such that each of the {\it Reifenberg sets} $\cR_{ε,r}(Y)$ is bi-Hölder homeomorphic to a manifold. It has been an ongoing question as to the bi-Lipschitz regularity the Reifenberg sets. Our results have two parts in this paper. First we show that each of the sets $\cR_{ε,r}(Y)$ are bi-Lipschitz embeddable into Euclidean space. Conversely, we show the bi-Lipschitz nature of the embedding is sharp. In fact, we construct a limit space $Y$ which is even uniformly Reifenberg, that is, not only is each tangent cone of $Y$ isometric to $\RR^n$ but convergence to the tangent cones is at a uniform rate in $Y$, such that there exists no $C^{1,β}$ embeddings of $Y$ into Euclidean space for any $β>0$. Further, despite the strong tangential regularity of $Y$, there exists a point $y\in Y$ such that every pair of minimizing geodesics beginning at $y$ branches to any order at $y$. More specifically, given {\it any} two unit speed minimizing geodesics $γ_1$, $γ_2$ beginning at $y$ and {\it any} $0\leq θ\leq π$, there exists a sequence $t_i\to 0$ such that the angle $\angle γ_1(t_i)yγ_2(t_i)$ converges to $θ$.

math.DG↗

arXiv2011

Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It reveals new, previously unknown, properties that all generalized spaces with a lower Ricci curvature bound must have and it has a number of applications. This new kind of estimate asserts that the geometry of small balls along any minimizing geodesic changes in a Hölder continuous way with a constant depending on the lower bound for the Ricci curvature, the dimension of the manifold, and the distance to the end points of the geodesic. We give examples that show that the Hölder exponent, along with essentially all the other consequences that we show follow from this estimate, are sharp. The unified theme for all of these applications is convexity. Among the applications is that the regular set is convex for any non-collapsed limit of Einstein metrics. In the general case of potentially collapsed limits of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and $a.e.$ convex, that is almost every pair of points can be connected by a minimizing geodesic whose interior is contained in the regular set. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group; the key point for this is to rule out small subgroups. The other asserts that the dimension of any limit space is the same everywhere. Finally, we show that a Reifenberg type property holds for collapsed limits and discuss why this indicate further regularity of manifolds and spaces with Ricci curvature bounds.

math.DG↗

arXiv2011

Noncompact Shrinking 4-Solitons with Nonnegative Curvature

We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc>= 0.

math.DG↗

arXiv2009

Geometric Structures of Collapsing Riemannian Manifolds II: N*-bundles and Almost Ricci Flat Spaces

In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the M_{i} themselves, this structure lives over X and allows for a convenient notion of global convergence as well as the appropriate background structure for doing analysis on X. This structure is new even in the case of uniformly bounded curvature and as an application we give a generalization of Gromov's Almost Flat Theorem and prove new Ricci pinching theorems which extend those known in the noncollapsed setting. There are also interesting topological consequences to the structure.

math.DG↗

arXiv2008

Geometric Structures of Collapsing Riemannian Manifolds I

Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional Riemannian Manifolds with uniform bounds on the sectional curvatures and let (X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the Gromov-Hausdorff sense. Let O \subseteq X be the set of points x \in X such that there exists a neighborhood of x which is isometric to an open set in a Riemannian orbifold and let B = O^c be the complement set. Then we have the sharp estimates dim_Haus(B) \leq min{n-5, dim_Haus(X)-3}, and further for arbitrary x \in X we have that x \in O iff a neighborhood of x has bounded Alexandroff curvature. In particular, if n \leq 4 then B is empty and (X,d) is a Riemannian orbifold. Our main application is to prove that a collapsed limit of Einstein four manifolds has a smooth Riemannian orbifold structure away from a finite number of points, and that near these points the curvatures has a -dist^{-2} lower bound.

math.DG↗

arXiv2006

Some Geometry and Analysis on Ricci Solitons

The Bakry-Emery Ricci tensor of a metric-measure space (M,g,e^{-f}dv_{g}) plays an important role in both geometric measure theory and the study of Hamilton's Ricci flow. Under a uniform positivity condition on this tensor and with bounded Ricci curvature we show the underlying space has finite f-volume. As a consequence such manifolds, including shrinking Ricci solitons, have finite fundamental group. The analysis can be extended to classify shrinking solitons under convexity or concavity assumptions on the measure function.

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