We study that the n−graphs defining by smooth map f : Ω ⊂ R n → R m , m ≥ 2, in R m+n of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates sup DR(x)

The introductory part will feature: rate of approximation of real numbers by rationals; theorems of Kronecker, Dirichlet, Liouville, Borel-Cantelli; connections with dynamical systems: circle rotations, hyperbolic flow in the space of lattices, geodesic flow on the modular survace, Gauss map (continued fractions); ergodicity, unique ergodicity, mixing, applications to uniform distribution of se...

In this paper, we prove the following two results: First, we study a class of conformally invariant operators P and their related conformally invariant curvatures Q on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, Q-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaf...

We consider two aspects of Kronecker coefficients in the directions of representation theory and combinatorics. We consider a conjecture of Jan Saxl stating that the tensor square of the Sn-irreducible representation indexed by the staircase partition contains every irreducible representation of Sn. We present a sufficient condition allowing to determine whether an irreducible representation is...

In this paper we discuss the problem of isometric embedding of the surface of a rapidly rotating black hole in a flat space. It is well known that intrinsically defined Riemannian manifolds can be isometrically embedded in a flat space. According to the Cartan-Janet [1, 2] theorem, every analytic Riemannian manifold of dimension n can be locally real analytically isometrically embedded into E w...

A Gauss Equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature ≤ A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature ≤ K +A. Sharp bounds on injectivity radii of subspaces, new even in the Riemannian case, are derived.

Let F : Σ n × [0, T) → R n+m be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps γ : (Σ n , g t) → G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g t. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m)...

In this paper, we prove the following two results: First, we study a class of conformally invariant operators P and their related conformally invariant curvatures Q on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, Q-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaf...

1. Introduction. Perhaps the most significant aspect of differential geometry is that which deals with the relationship between the curvature properties of a Riemannian manifold M and its topological structure. One of the beautiful results in this connection is the (generalized) Gauss-Bonnet theorem which relates the curvature of compact and oriented even-dimensional manifolds with an important...

Second-order optimization methods have the ability to accelerate convergence by modifying gradient through curvature matrix. There been many attempts use second-order for training deep neural networks. In this work, inspired diagonal approximations and factored such as Kronecker-factored Approximate Curvature (KFAC), we propose a new approximation Fisher information matrix (FIM) called Trace-re...