Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA sufficient condition for a hypersurface to be isoparametric
177
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $mathcal{A}$ has $n$ distinct eigenvalues, and $mathrm{tr}(mathcal{A}^k)$ are constants for $k=1,cdots, n-1$. We show that all the eigenvalues of $mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito cite{dB90} to higher dimensions. As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric if one takes $mathfrak{a}$ above to be the second fundamental form, giving affirmative evidence to Cherns conjecture.
rate research
Read More
We generalize the concept of the group determinant and prove a necessary and sufficient novel condition for a subset to be a subgroup. This development is based on the group determinant work by Edward Formanek, David Sibley, and Richard Mansfield, where they show that two groups with the same group determinant are isomorphic. The derived condition leads to a generalization of this result.
Several counterexample models to the Nelson-Seiberg theorem have been discovered in previous literature, with generic superpotentials respecting the R-symmetry and non-generic R-charge assignments for chiral fields. This work present a sufficient condition for such counterexample models: The number of R-charge 2 fields, which is greater than the number of R-charge 0 fields, must be less than or equal to the number of R-charge 0 fields plus the number of independent field pairs with opposite R-charges and satisfying some extra requirements. We give a correct count of such field pairs when there are multiple field pairs with degenerated R-charges. These models give supersymmetric vacua with spontaneous R-symmetry breaking, thus are counterexamples to both the Nelson-Seiberg theorem and its extensions.
Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Cherns conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.
Let $D$ be a disk in $mathbb{R}^n$ and $fin C^{r+2}(D, mathbb{R}^k)$. We deal with the problem of the algebraic approximation of the set $j^{r}f^{-1}(W)$ consisting of the set of points in the disk $D$ where the $r$-th jet extension of $f$ meets a given semialgebraic set $Wsubset J^{r}(D, mathbb{R}^k).$ Examples of sets arising in this way are the zero set of $f$, or the set of its critical points. Under some transversality conditions, we prove that $f$ can be approximated with a polynomial map $p:Dto mathbb{R}^k$ such that the corresponding singularity is diffeomorphic to the original one, and such that the degree of this polynomial map can be controlled by the $C^{r+2}$ data of $f$. More precisely, begin{equation} text{deg}(p)le Oleft(frac{|f|_{C^{r+2}(D, mathbb{R}^k)}}{mathrm{dist}_{C^{r+1}}(f, Delta_W)}right), end{equation} where $Delta_W$ is the set of maps whose $r$-th jet extension is not transverse to $W$. The estimate on the degree of $p$ implies an estimate on the Betti numbers of the singularity, however, using more refined tools, we prove independently a similar estimate, but involving only the $C^{r+1}$ data of $f$. These results specialize to the case of zero sets of $fin C^{2}(D, mathbb{R})$, and give a way to approximate a smooth hypersurface defined by the equation $f=0$ with an algebraic one, with controlled degree (from which the title of the paper). In particular, we show that a compact hypersurface $Zsubset Dsubset mathbb{R}^n$ with positive reach $rho(Z)>0$ is isotopic to the zero set in $D$ of a polynomial $p$ of degree begin{equation} text{deg}(p)leq c(D)cdot 2 left(1+frac{1}{rho(Z)}+frac{5n}{rho(Z)^2}right),end{equation} where $c(D)>0$ is a constant depending on the size of the disk $D$ (and in particular on the diameter of $Z$).
Based on the general form of entanglement witnesses constructed from separable states, we first show a sufficient condition of violating the structural physical approximation (SPA) conjecture [Phys. Rev. A 78, 062105 (2008)]. Then we discuss the SPA conjecture for decomposable entanglement witnesses. Moreover, we make geometric illustrations of the connection between entanglement witnesses and the sets of quantum states, separable states, and entangled states comparing with planes and vectors in Euclidean space.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
