An official website of the United States government
Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher.
Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?
Some links on this page may take you to non-federal websites. Their policies may differ from this site.
-
We show that a complete [Formula: see text]-dimensional Riemannian manifold [Formula: see text] with finitely generated first homology has macroscopic dimension [Formula: see text] if it satisfies the following “macroscopic curvature” assumptions: every ball of radius [Formula: see text] in [Formula: see text] has volume at most [Formula: see text], and every loop in every ball of radius [Formula: see text] in [Formula: see text] is null-homologous in the concentric ball of radius [Formula: see text].more » « less
-
Abstract We study the degree of an L-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $$X_k$$ is the connected sum of k copies of $$\mathbb CP^2$$for$$k \ge 4$$, then we prove that the maximum degree of an L-Lipschitz self-map of $$X_k$$ is between $$C_1 L^4 (\log L)^{-4}$$ and $$C_2 L^4 (\log L)^{-1/2}$$. More generally, we divide simply connected manifolds into three topological types with three different behaviors. Each type is defined by purely topological criteria. For scalable simply connected n-manifolds, the maximal degree is $$\sim L^n$$. For formal but nonscalable simply connectedn-manifolds, the maximal degree grows roughly like $$L^n (\log L)^{-\theta (1)}$$. And for nonformal simply connected n-manifolds, the maximal degree is bounded by $$L^\alpha $$ for some $$\alpha < n$$.more » « less
-
We prove sharp bounds for the size of superlevel sets\{x \in \mathbb{R}^2\colon |f(x)| >\alpha\}, where\alpha>0andf\colon\mathbb{R}^2\to\mathbb{C}is a Schwartz function with Fourier transform supported in anR^{-1}-neighborhood of the truncated parabola\mathbb{P}^1. These estimates imply the small cap decoupling theorem for\mathbb{P}^1of Demeter, Guth, and Wang (2020) and the canonical decoupling theorem for\mathbb{P}^1of Bourgain and Demeter (2015). New(\ell^q,L^p)small cap decoupling inequalities also follow from our sharp level set estimates.more » « less
Full Text Available