arXiv research

New inequalities for austere submanifolds established.

Survey of recent austere submanifold results.

New examples of austere submanifolds and hypersurfaces with specific curvature properties.

Classifies twisted-austere 3-folds in Euclidean space.

The paper introduces austere and arid submanifolds in Hilbert spaces.

We introduce the notion of a weakly reflective submanifold, which is an austere submanifold with a certain global condition, and study its fundamental properties. Using these, we determine weakly reflective orbits and austere orbits of s-representations.

Harvey-Lawson and Anciaux introduced the notion of austere submanifolds in pseudo-Riemannian geometry. We give an equivalent condition for an orbit of the isotropy representations for semisimple pseudo-Riemannian symmetric space to be an austere submanifold in a pseudo-sphere in terms of restricted root system theory w…

For an arbitrary submanifold $M \subset \mathbb{C}P^n$ we determine conditions under which it is austere, i.e., the normal bundle of $M$ is special Lagrangian with respect to Stenzel's Ricci-flat Kähler metric on $T\mathbb{C}P^n$. We also classify austere surfaces in $\mathbb{C}P^n$.

Austere submanifolds of Euclidean space were introduced in 1982 by Harvey and Lawson in their foundational work on calibrated geometries. In general, the austerity condition is much stronger than minimality since it express that the nonzero eigenvalues of the shape operator of the submanifold appear in opposite pairs f…

Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of 4-dimensional austere submanifolds were classified by Bryant, into three types which we label A, B, and C. In this paper, we s…

The paper studies curvatures and austere properties of orbits in symmetric spaces.

We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the…

We give examples of certain kind of minimal orbits of Hermann actions and discuss whether each of the examples is austere.

The main result of this paper was already obtained in the paper `Some Remarks on the Geometry of Austere Manifolds', by Robert L. Bryant.

New examples of isoparametric families on non-compact symmetric spaces.

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds…

We give a geometric characterization of certain hypersurfaces of cohomogeneity one in the complex projective and hyperbolic planes. We also obtain some partial classifications of austere hypersurfaces and of Levi-flat hypersurfaces with constant mean curvature in these spaces.

This summarizes the study of the financial and economic crisis in Europe. The starting questions were: 1) Why do we have a crisis? Unde venis? 2) What will be the outcome? Quo vadis? Here is the reasoning which touches many areas, ranging from financial to politics and from psychology and economy.

Stark hypersurfaces are a special class of austere hypersurface in ${\mathbb C}P^n$ where the shape operator is compatible with the $CR$-structure. In this paper, the possible shape operators for stark hypersurfaces are completely determined, and stark hypersurfaces in ${\mathbb C}P^2$ are constructed as integrals of a…

Wintgen ideal submanifolds in space forms are those ones attaining equality at every point in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the normal scalar curvature. This property is conformal invariant; hence we study them in the framework of Moebius geometry, and restrict…

In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total $2p$-th mean curvature functional $\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold $N^{n+m}$ for $p=0,1,...,[\frac{n}{2}]$. As an example, we prove that closed complex submanifolds in compl…

First-best climate policy is a uniform carbon tax which gradually rises over time. Civil servants have complicated climate policy to expand bureaucracies, politicians to create rents. Environmentalists have exaggerated climate change to gain influence, other activists have joined the climate bandwagon. Opponents to cli…

We show that there is no uniform upper bound on |Out(Aut(A))| when A ranges over all right-angled Artin groups. This is in contrast with the cases where A is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed that Out(Aut(F_n)) = 1, while Hua-Reiner showed |Out(Aut(Z^n)| = |Out(GL(n,Z))| < 5. We…

Study on $G_{2}$ actions on symmetric spaces, focusing on orbit properties.

Can we make Bayesian posterior MCMC sampling more efficient when faced with very large datasets? We argue that computing the likelihood for N datapoints in the Metropolis-Hastings (MH) test to reach a single binary decision is computationally inefficient. We introduce an approximate MH rule based on a sequential hypoth…

The seriousness of the current crisis urgently demands new economic thinking that breaks the austerity vs. deficit spending circle in economic policy. The core tenet of the paper is that the most important problems that natural and social science are facing today are inverse problems, and that a new approach that goes …

New method finds sparse networks without labels, improving performance.

This paper investigates the relevance of the No-Ponzi game condition for public debt (i.e. the public debt growth rate has to be lower than the real interest rate, a necessary assumption for Ricardian equivalence) and of the transversality condition for the GDP growth rate (i.e. the GDP growth rate has to be lower than…

In this paper we investigate $m$-dimensional complete minimal submanifolds in Euclidean spheres with index of relative nullity at least $m-2$ at any point. These are austere submanifolds in the sense of Harvey and Lawson \cite{harvey} and were initially studied by Bryant \cite{br}. For any dimension and codimension the…

Extends PF submanifold results and connects Kac-Moody spaces.

Since beginning of the 2008 financial crisis almost half a trillion euros have been spent to financially assist EU member states in taxpayer-funded bail-outs. These crisis resolutions are often accompanied by austerity programs causing political and social friction on both domestic and international levels. The questio…

In the panoply of pattern classification techniques, few enjoy the intuitive appeal and simplicity of the nearest neighbor rule: given a set of samples in some metric domain space whose value under some function is known, we estimate the function anywhere in the domain by giving the value of the nearest sample per the …

During the last two years, Europe has been facing a debt crisis, and Greece has been at its center. In response to the crisis, drastic actions have been taken, including the halving of Greek debt. Policy makers acted because interest rates for sovereign debt increased dramatically. High interest rates imply that defaul…