اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA topological pinching for the injectivity radius of a compact surface in S^3 and in H^3
106
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
It is given a topological pinching for the injectivity radius of a compact embedded surface either in the sphere or in the hyperbolic space
قيم البحث
اقرأ أيضاً
We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of $S^3$; 2) in
$H^3$, the smaller volume is at least 85% that of the larger. A balancing argument and asymptotic analysis reduce the problem in $S^3$ and $H^3$ to some computer checking. The computer analysis has been designed and fully implemented for both spaces.
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $operatorname{conv}(p)ge min{fr
ac{1}{2}operatorname{inj}(p),operatorname{foc}(B_{operatorname{inj}(p)}(p))}$, where $operatorname{inj}(p)$ is the injectivity radius of $p$ and $operatorname{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $operatorname{inj}(q)ge min{operatorname{inj}(p), operatorname{conj}(q)}-d(p,q),$ where $operatorname{conj}(q)$ is the conjugate radius of $q$; 3) for any $0<r<min{operatorname{inj}(p),frac{1}{2}operatorname{conj}(B_{operatorname{inj}(p)}(p))}$, any (not necessarily minimizing) geodesic in $B_r(p)$ has length $le 2r$. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.
Let $M$ be a compact constant mean curvature surface either in $mathbb{S}^3$ or $mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a comparison theorem b
etween the spectrum of the Jacobi operator of $M$ and those of Hodge Laplacian of $1$-forms on $M$.
We introduce three non-trivial 2-cocycles $c_k$, k=0,1,2, on the Lie algebra $S^3H=Map(S^3,H)$ with the aid of the corresponding basis vector fields on $S^3$, and extend them to 2-cocycles on the Lie algebra $S^3gl(n,H)=S^3H otimes gl(n,C)$. Then we
have the corresponding central extension $S^3gl(n,H)oplus oplus_k (Ca_k)$. As a subalgebra of $S^3H$ we have the algebra $C[phi]$ of the Laurent polynomial spinors on $S^3$. Then we have a Lie subalgebra $hat{gl}(n, H)=C[phi] otimes gl(n, C)$ of $S^3gl(n,H)$, as well as its central extension by the 2-cocycles ${c_k}$ and the Euler vector field $d$: $hat{gl}=hat{gl}(n, H) oplus oplus_k(Ca_k)oplus Cd$ . The Lie algebra $hat{sl}(n,H)$ is defined as a Lie subalgebra of $hat{gl}(n,H)$ generated by $C[phi]otimes sl(n,C))$. We have the corresponding central extension of $hat{sl}(n,H)$ by the 2-cocycles ${c_k}$ and the derivation $d$, which becomes a Lie subalgebra $hat{sl}$ of $hat{gl}$. Let $h_0$ be a Cartan subalgebra of $sl(n,C)$ and $hat{h}=h_0 oplus oplus_k(Ca_k)oplus Cd$. The root space decomposition of the $ad(hat{h})$-representation of $hat{sl}$ is obtained. The set of roots is $Delta ={ m/2 delta + alpha ; alpha in Delta_0, m in Z} bigcup {m/2 delta ; m in Z }$ . And the root spaces are $hat{g}_{m/2 delta+ alpha}= C[phi ;m] otimes g_{alpha}$, for $alpha eq 0$ , $hat{g}_{m/2 delta}= C[phi ;m] otimes h_0$, for $m eq 0$, and $hat{g}_{0 delta}= hat{h}$, where $C[phi ;m]$ is the subspace with the homogeneous degree m. The Chevalley generators of $hat{sl}$ are given.
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an iso
metry in the following sense: it preserves the metric coefficients introduced previously by Bobenko and Suris for isothermic nets. Exactly as in the smooth case, this is a correspondence between nets with the same Lax matrices, and the immersion formulas also coincide with the smooth case.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
