Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWhat is the Degree of a Smooth Hypersurface?
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $Z$).
rate research
Read More
Let $X$ be a compact manifold, $G$ a Lie group, $P to X$ a principal $G$-bundle, and $mathcal{B}_P$ the infinite-dimensional moduli space of connections on $P$ modulo gauge. For a real elliptic operator $E_bullet$ we previously studied orientations on the real determinant line bundle over $mathcal{B}_P$. These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson. Here we consider complex elliptic operators $F_bullet$ and introduce the idea of spin structures, square roots of the complex determinant line bundle of $F_bullet$. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures. Our main result identifies spin structures on $X$ with orientations on $X times S^1$. Thus, if $P to X$ and $Q to X times S^1$ are principal $G$-bundles with $Q|_{Xtimes{1}} cong P$, we relate spin structures on $(mathcal{B}_P,F_bullet)$ to orientations on $(mathcal{B}_Q,E_bullet)$ for a certain class of operators $F_bullet$ on $X$ and $E_bullet$ on $Xtimes S^1$. Combined with arXiv:1811.02405, we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups $G=U(m), SU(m)$. In a sequel arXiv:2001.00113 we apply this to define canonical orientation data for all Calabi-Yau 3-folds $X$ over the complex numbers, as in Kontsevich-Soibelman arXiv:0811.2435, solving a long-standing problem in Donaldson-Thomas theory.
In the present paper, we revisit the rigidity of hypersurfaces in Euclidean space. We highlight Darboux equation and give new proof of rigidity of hypersurfaces by energy method and maximal principle.
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.
The degree of mobility of a (pseudo-Riemannian) Kahler metric is the dimension of the space of metrics h-projectively equivalent to it. We prove that a metric on a closed connected manifold can not have the degree of mobility $ge 3$ unless it is essentially the Fubini-Study metric, or the h-projective equivalence is actually the affine equivalence. As the main application we prove an important special case of the classical conjecture attributed to Obata and Yano, stating that a closed manifold admitting an essential group of h-projective transformations is $(CP(n), g_{Fubini-Study})$ (up to a multiplication of the metric by a constant). An additional result is the generalization of a certain result of Tanno 1978 for the pseudo-Riemannian situation.
We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold for a large class of operators containing in particular the p-Laplacian and the minimal graph operator.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
