Four-Manifolds, Curvature Bounds, and Convex Geometry

Abstract In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also explore functional inequalities derived from the comparison of the dilation profiles under the nonnegative weighted Ricci curvature. In particular, we show several functional inequalities related to various entropies.

Download Full-text

Constructing symplectic manifolds

10.1093/oso/9780198794899.003.0008 ◽

2017 ◽

Author(s):

Dusa McDuff ◽

Dietmar Salamon

Keyword(s):

Final Section ◽

Symplectic Manifolds ◽

Homotopy Equivalence ◽

Codimension Two ◽

Open Manifolds ◽

Connected Sums ◽

Symplectic Forms ◽

Ambient Manifold ◽

Blowing Up ◽

Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

Download Full-text

Quantum Polar Duality and the Symplectic Camel: A New Geometric Approach to Quantization

Foundations of Physics ◽

10.1007/s10701-021-00465-6 ◽

2021 ◽

Vol 51 (3) ◽

Author(s):

Maurice A. de Gosson

Keyword(s):

Uncertainty Principle ◽

Convex Geometry ◽

Geometric Approach ◽

Point Of View ◽

Reconstruction Problem ◽

Orthogonal Projections ◽

Topological Version ◽

Mahler Conjecture ◽

Quantum Indeterminacy ◽

Dual Quantum

AbstractWe define and study the notion of quantum polarity, which is a kind of geometric Fourier transform between sets of positions and sets of momenta. Extending previous work of ours, we show that the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what we call a dual quantum pair. We thereafter show that quantum polarity allows solving the Pauli reconstruction problem for Gaussian wavefunctions. The notion of quantum polarity exhibits a strong interplay between the uncertainty principle and symplectic and convex geometry and our approach could therefore pave the way for a geometric and topological version of quantum indeterminacy. We relate our results to the Blaschke–Santaló inequality and to the Mahler conjecture. We also discuss the Hardy uncertainty principle and the less-known Donoho–Stark principle from the point of view of quantum polarity.

Download Full-text

Continuous Maps from Spheres Converging to Boundaries of Convex Hulls

Forum of Mathematics Sigma ◽

10.1017/fms.2021.10 ◽

2021 ◽

Vol 9 ◽

Author(s):

Joseph Malkoun ◽

Peter J. Olver

Keyword(s):

Convex Hull ◽

Convex Geometry ◽

Gauss Map ◽

Parameter Family ◽

Convex Hulls ◽

Dimensional Sphere ◽

Continuous Maps ◽

Dimensional Boundary

Abstract Given n distinct points $\mathbf {x}_1, \ldots , \mathbf {x}_n$ in $\mathbb {R}^d$ , let K denote their convex hull, which we assume to be d-dimensional, and $B = \partial K $ its $(d-1)$ -dimensional boundary. We construct an explicit, easily computable one-parameter family of continuous maps $\mathbf {f}_{\varepsilon } \colon \mathbb {S}^{d-1} \to K$ which, for $\varepsilon> 0$ , are defined on the $(d-1)$ -dimensional sphere, and whose images $\mathbf {f}_{\varepsilon }({\mathbb {S}^{d-1}})$ are codimension $1$ submanifolds contained in the interior of K. Moreover, as the parameter $\varepsilon $ goes to $0^+$ , the images $\mathbf {f}_{\varepsilon } ({\mathbb {S}^{d-1}})$ converge, as sets, to the boundary B of the convex hull. We prove this theorem using techniques from convex geometry of (spherical) polytopes and set-valued homology. We further establish an interesting relationship with the Gauss map of the polytope B, appropriately defined. Several computer plots illustrating these results are included.

Download Full-text

Translating between the representations of a ranked convex geometry

Discrete Mathematics ◽

10.1016/j.disc.2021.112399 ◽

2021 ◽

Vol 344 (7) ◽

pp. 112399

Author(s):

Oscar Defrain ◽

Lhouari Nourine ◽

Simon Vilmin

Keyword(s):

Convex Geometry

Download Full-text

Remarks on Manifolds with Two-Sided Curvature Bounds

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0122 ◽

2021 ◽

Vol 9 (1) ◽

pp. 53-64

Author(s):

Vitali Kapovitch ◽

Alexander Lytchak

Keyword(s):

Distance Functions ◽

Cut Locus ◽

Bounded Curvature ◽

Curvature Bounds ◽

Positive Reach

Abstract We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

Download Full-text

Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Analysis ◽

10.1515/anly-2020-0035 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Siran Li

Keyword(s):

Banach Space ◽

Discrete Geometry ◽

Thin Shell ◽

Euclidean Geometry ◽

Convex Geometry ◽

Regularity Theory ◽

Elliptic Partial Differential Equations ◽

High Dimensional ◽

Space Theory ◽

Elementary Calculus

AbstractIt is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in \mathbb{R}^{n} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n\nearrow\infty. Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

Download Full-text