Aaron Naber
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The organizations/institutes from the scholar's latest article
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Aaron Naber has a total of 2 co-authors. Here are the top co-authors by number of shared publications:
- Tobias Holck Colding
Dept. of Math., MIT, Cambridge, USA
3 shared publications - Hans-Joachim Hein
Mathematisches Institut, Universität Münster, Münster, Germany
2 shared publications
Latest Publications
- 26 July 2013
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 67 (9) , 1543-1561
New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow
In this note, we prove an ɛ‐regularity theorem for the Ricci flow. Let (Mn,g(t)) with t ∊ [−T,0] be a Ricci flow, and let Hx0(y,s) be the conjugate heat kernel centered at some point (x0,0) in the final time slice. By substituting Hx0(−,s) into Perelman's W‐functional, we obtain a monotone quantity Wx0(s) that we refer to as the pointed entropy. This satisfies Wx0(s) ≤ 0, and Wx0(s) = 0 if and only if (Mn,g(t)) is isometric to the trivial flow on Rn. Then our main theorem asserts the following: There exists ɛ > 0, depending only on T and on lower scalar curvature and μ‐entropy bounds for the initial slice (Mn,g(−T)) such that Wx0(s) ≥ −ɛ implies |Rm| ≤ r−2 on Pɛ r(x0,0), where r2 ≡ |s| and Pρ(x,t) ≡ Bρ(x,t) × (t−ρ2,t] is our notation for parabolic balls.The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s‐average of Wx(s). To accomplish this, we require a new log‐Sobolev inequality. Perelman's work implies that the metric measure spaces (Mn,g(t),dvolg(t)) satisfy a log‐Sobolev; we show that this is also true for the heat kernel weighted spaces (Mn,g(t),Hx0(−,t)dvolg(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 certain average Gaussian upper bounds on the conjugate heat kernel.
- 26 July 2013
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 67 (9) , 1543-1561
New Logarithmic Sobolev Inequalities and an ɛ‐Regularity Theorem for the Ricci Flow
In this note, we prove an ɛ‐regularity theorem for the Ricci flow. Let (Mn,g(t)) with t ∊ [−T,0] be a Ricci flow, and let Hx0(y,s) be the conjugate heat kernel centered at some point (x0,0) in the final time slice. By substituting Hx0(−,s) into Perelman's W‐functional, we obtain a monotone quantity Wx0(s) that we refer to as the pointed entropy. This satisfies Wx0(s) ≤ 0, and Wx0(s) = 0 if and only if (Mn,g(t)) is isometric to the trivial flow on Rn. Then our main theorem asserts the following: There exists ɛ > 0, depending only on T and on lower scalar curvature and μ‐entropy bounds for the initial slice (Mn,g(−T)) such that Wx0(s) ≥ −ɛ implies |Rm| ≤ r−2 on Pɛ r(x0,0), where r2 ≡ |s| and Pρ(x,t) ≡ Bρ(x,t) × (t−ρ2,t] is our notation for parabolic balls.The main technical challenge of the theorem is to prove an effective Lipschitz bound in x for the s‐average of Wx(s). To accomplish this, we require a new log‐Sobolev inequality. Perelman's work implies that the metric measure spaces (Mn,g(t),dvolg(t)) satisfy a log‐Sobolev; we show that this is also true for the heat kernel weighted spaces (Mn,g(t),Hx0(−,t)dvolg(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 certain average Gaussian upper bounds on the conjugate heat kernel.
- Open Access
- 26 January 2013
- Published by Springer Nature in Geometric and Functional Analysis
- Vol. 23 (1) , 134-148
Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications
- 26 January 2013
- Published by Springer Nature in Geometric and Functional Analysis
- Vol. 23 (1) , 134-148
Open AccessCharacterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications
- Open Access
New logarithmic Sobolev inequalities and an ε-regularity theorem for the Ricci flow
In this note we prove a new \epsilon-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 \epsilon>0, depending only on T and on lower scalar curvature and \mu-entropy bounds for (M,g(-T)), such that W_{x_0}(s) > -\epsilon implies |Rm|< r^{-2} on P_{\epsilon r}(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.
Open AccessNew logarithmic Sobolev inequalities and an ε-regularity theorem for the Ricci flow
In this note we prove a new \epsilon-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 \epsilon>0, depending only on T and on lower scalar curvature and \mu-entropy bounds for (M,g(-T)), such that W_{x_0}(s) > -\epsilon implies |Rm|< r^{-2} on P_{\epsilon r}(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.
- Open Access
Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications
Consider a limit space $(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_\alpha^n$ 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 $\bar\Omega_{Y,p}\subseteq\cM_{GH}$ 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 $\bar\Omega_{Y,p}$. In particular, we give necessary and sufficient conditions for an open smooth family $\Omega\equiv (X_s,g_s)$ of closed manifolds to satisfy $\bar\Omega =\bar\Omega_{Y,p}$ for {\it some} limit $Y$ and point $p\in Y$ as above, where $\bar\Omega$ is the closure of $\Omega$ 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 $\Sn^4$.Open AccessCharacterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications
Consider a limit space $(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_\alpha^n$ 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 $\bar\Omega_{Y,p}\subseteq\cM_{GH}$ 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 $\bar\Omega_{Y,p}$. In particular, we give necessary and sufficient conditions for an open smooth family $\Omega\equiv (X_s,g_s)$ of closed manifolds to satisfy $\bar\Omega =\bar\Omega_{Y,p}$ for {\it some} limit $Y$ and point $p\in Y$ as above, where $\bar\Omega$ is the closure of $\Omega$ 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 $\Sn^4$. - Open Access
Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces
We study here limit spaces $(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_\alpha$ 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 $\cap_\epsilon \cup_r\cR_{\epsilon,r}(Y)=\cR(Y)$ such that each of the {\it Reifenberg sets} $\cR_{\epsilon,r}(Y)$ is bi-H\"older 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 $\cR_{\epsilon,r}(Y)$ 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 $C^{1,\beta}$ embeddings of $Y$ into Euclidean space for any $\beta>0$. 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 $\gamma_1$, $\gamma_2$ beginning at $y$ and {\it any} $0\leq \theta\leq \pi$, there exists a sequence $t_i\to 0$ such that the angle $\angle \gamma_1(t_i)y\gamma_2(t_i)$ converges to $\theta$.Open AccessLower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces
We study here limit spaces $(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_\alpha$ 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 $\cap_\epsilon \cup_r\cR_{\epsilon,r}(Y)=\cR(Y)$ such that each of the {\it Reifenberg sets} $\cR_{\epsilon,r}(Y)$ is bi-H\"older 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 $\cR_{\epsilon,r}(Y)$ 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 $C^{1,\beta}$ embeddings of $Y$ into Euclidean space for any $\beta>0$. 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 $\gamma_1$, $\gamma_2$ beginning at $y$ and {\it any} $0\leq \theta\leq \pi$, there exists a sequence $t_i\to 0$ such that the angle $\angle \gamma_1(t_i)y\gamma_2(t_i)$ converges to $\theta$.