We introduce the notion of the projective linking number LinkP(Γ, Z) of a compact oriented real submanifold Γ of dimension 2p − 1 in complex projective n-space P with an algebraic subvariety Z ⊂ P − Γ of codimension p. This notion is related to projective winding numbers and quasi-plurisubharmonic functions, and it generalizes directly from P to any projective manifold. Part 1 of this paper est...

Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much ...

A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any two distinct points are not parallel. We show that if an oriented closed manifold has a non-zero Euler characteristic χ then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ/4. We give a version of this result for immersed surf...

The main purpose of this paper is to give a construction of certain " mirror dual " Calabi-Yau submanifolds inside of a G2 manifold. More specifically, we explain how to assign a G2 manifold (M, ϕ, Λ), with the calibration 3-form ϕ and an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M. These forms can then be used to define different complex and symple...

The projection of a compact oriented submanifold M ⊂ R n+1 on a hyperplane P n can fail to bound any region in P . We call this “projecting to zero.” Example: The equatorial S ⊂ S ⊂ R projects to zero in any plane containing the x3-axis. Using currents to make this precise, we show: A lipschitz (homology) (n − 1)-sphere embedded in a compact, strictly convex hypersurface cannot project to zero ...

This article introduces a universal moduli space for the set whose archetypal element is a pair that consists of a metric and second fundamental form from a compact, oriented, positive genus minimal surface in some hyperbolic 3–manifold. This moduli space is a smooth, finite dimensional manifold with canonical maps to both the cotangent bundle of the Teichmüller space and the space of SO3(C) re...

We give asymptotically tight estimate of tangent space variation on a Riemannian submanifold of Euclidean space with respect to the local feature size of the submanifold. We show that the result is a consequence of structural properties of local feature size function of the Riemannian submanifold.

The main purpose of this paper is to give a construction of certain “mirror dual” Calabi-Yau submanifolds inside of a G2 manifold. More specifically, we explain how to assign a G2 manifold (M, φ,Λ), with the calibration 3-form φ and an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2 and 3-forms of M . These forms can then be used to define different complex and symplect...