Cambridge Journal of Mathematics最新文献_第4页

Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers 推广线性化加倍方法，I：一般理论与新的极小曲面和自收缩器

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2020-01-13 DOI: 10.4310/cjm.2023.v11.n2.a1

N. Kapouleas, Peter J. McGrath

{"title":"Generalizing the Linearized Doubling approach, I: General theory and new minimal surfaces and self-shrinkers","authors":"N. Kapouleas, Peter J. McGrath","doi":"10.4310/cjm.2023.v11.n2.a1","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n2.a1","url":null,"abstract":"In Part I of this article we generalize the Linearized Doubling (LD) approach, introduced in earlier work by NK, by proving a general theorem stating that if $Sigma$ is a closed minimal surface embedded in a Riemannian three-manifold $(N,g)$ and its Jacobi operator has trivial kernel, then given a suitable family of LD solutions on $Sigma$, a minimal surface $breve{M}$ resembling two copies of $Sigma$ joined by many small catenoidal bridges can be constructed by PDE gluing methods. (An LD solution $varphi$ on $Sigma$ is a singular solution of the Jacobi equation with logarithmic singularities which in the construction are replaced by catenoidal bridges.) We also determine the first nontrivial term in the expansion for the area $|breve{M}|$ of $breve{M}$ in terms of the sizes of its catenoidal bridges and confirm that it is negative; $|breve{M}|<2 | Sigma|$ follows. We demonstrate the applicability of the theorem by first constructing new doublings of the Clifford torus. We then construct in Part II families of LD solutions for general $(O(2)times mathbb{Z}_2)$-symmetric backgrounds $(Sigma, N,g)$. Combining with the theorem in Part I this implies the construction of new minimal doublings for such backgrounds. (Constructions for general backgrounds remain open.) This generalizes our earlier work for $Sigma=mathbb{S}^2 subset N=mathbb{S}^3$ providing new constructions even in that case. In Part III, applying the results of Parts I and II -- appropriately modified for the catenoid and the critical catenoid -- we construct new self-shrinkers of the mean curvature flow via doubling the spherical self-shrinker or the Angenent torus, new complete embedded minimal surfaces of finite total curvature in the Euclidean three-space via doubling the catenoid, and new free boundary minimal surfaces in the unit ball via doubling the critical catenoid.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2020-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41549152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 11

Non-concavity of the Robin ground state 罗宾基态的非凹性

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2020-01-01 DOI: 10.4310/cjm.2020.v8.n2.a1

B. Andrews, J. Clutterbuck, Daniel Hauer

引用次数: 9

On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices 关于确定性矩阵和iid-GUE矩阵中非交换多项式的算子范数

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-12-10 DOI: 10.4310/cjm.2022.v10.n1.a3

B. Collins, A. Guionnet, Félix Parraud

{"title":"On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices","authors":"B. Collins, A. Guionnet, Félix Parraud","doi":"10.4310/cjm.2022.v10.n1.a3","DOIUrl":"https://doi.org/10.4310/cjm.2022.v10.n1.a3","url":null,"abstract":"Let $X^N = (X_1^N,dots, X^N_d)$ be a d-tuple of $Ntimes N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $mathbb{M}_N(mathbb{C})otimes mathbb{M}_M(mathbb{C})$. Let $P$ be a self-adjoint non-commutative polynomial. A seminal work of Voiculescu shows that the empirical measure of the eigenvalues of $P(X^N)$ converges towards a deterministic measure defined thanks to free probability theory. Let now $f$ be a smooth function, the main technical result of this paper is a precise bound of the difference between the expectation of $$frac{1}{MN}text{Tr}left( f(P(X^Notimes I_M,Z^{NM})) right)$$ and its limit when $N$ goes to infinity. If $f$ is six times differentiable, we show that it is bounded by $M^2leftVert frightVert_{mathcal{C}^6}N^{-2}$. As a corollary we obtain a new proof of a result of Haagerup and Thorbjo rnsen, later developed by Male, which gives sufficient conditions for the operator norm of a polynomial evaluated in $(X^N,Z^{NM},{Z^{NM}}^*)$ to converge almost surely towards its free limit. Restricting ourselves to polynomials in independent GUE matrices, we give concentration estimates on the largest eingenvalue of these polynomials around their free limit. A direct consequence of these inequalities is that there exists some $beta>0$ such that for any $varepsilon_1<3+beta)^{-1}$ and $varepsilon_2<1/4$, almost surely for $N$ large enough, $$-frac{1}{N^{varepsilon_1}} leq | P(X^N)| - leftVert P(x)rightVert leq frac{1}{N^{varepsilon_2}}.$$ Finally if $X^N$ and $Y^{M_N}$ are independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any polynomial in $(X^Notimes I_{M_N},I_Notimes Y^{M_N})$ converges almost surely towards its free limit. This result is an improvement of a Theorem of Pisier, who was himself using estimates from Haagerup and Thorbjo rnsen, where $M_N$ had size $o(N^{1/4})$.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2019-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42366093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 19

On loop Deligne–Lusztig varieties of Coxeter-type for inner forms of $mathrm{GL}_n$ $ mathm {GL}_n$的内部形式的coxette类型的On循环delign - lusztig变体

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-11-08 DOI: 10.4310/cjm.2023.v11.n2.a2

C. Chan, A. Ivanov

引用次数: 4

On the stability of self-similar blow-up for $C^{1,alpha}$ solutions to the incompressible Euler equations on $mathbb{R}^3$ 关于$mathbb{R}^3$上不可压缩欧拉方程$C^{1， $ α}$解的自相似爆破的稳定性

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-10-30 DOI: 10.4310/cjm.2021.v9.n4.a4

T. Elgindi, T. Ghoul, N. Masmoudi

引用次数: 20

Crossed modular categories and the Verlinde formula for twisted conformal blocks 扭转共形块的交叉模范畴与Verlinde公式

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-09-24 DOI: 10.4310/cjm.2023.v11.n1.a2

Tanmay Deshpande, S. Mukhopadhyay

{"title":"Crossed modular categories and the Verlinde formula for twisted conformal blocks","authors":"Tanmay Deshpande, S. Mukhopadhyay","doi":"10.4310/cjm.2023.v11.n1.a2","DOIUrl":"https://doi.org/10.4310/cjm.2023.v11.n1.a2","url":null,"abstract":"In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Gamma$ and a positive integral level $ell$ under the assumption that \"$Gamma$ preserves a Borel\". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Gamma$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Gamma$-crossed modular functor and show that it is very closely related to the notion of a $Gamma$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $Gamma$-twisted conformal blocks associated with a twisted affine Lie algebra define a $Gamma$-crossed modular functor. Along the way, we prove equivalence between a $Gamma$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.","PeriodicalId":48573,"journal":{"name":"Cambridge Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2019-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47972892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 8

Equivariant Grothendieck–Riemann–Roch theorem via formal deformation theory 形式变形理论的等变Grothendieck–Riemann–Roch定理

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-06-01 DOI: 10.4310/cjm.2021.v9.n4.a1

G. Kondyrev, A. Prikhodko

引用次数: 4

Arithmetic subspaces of moduli spaces of rank one local systems 一阶局部系统模空间的算术子空间

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-02-08 DOI: 10.4310/cjm.2020.v8.n3.a1

H. Esnault, M. Kerz

引用次数: 9

Topological uniqueness for self-expanders of small entropy 小熵自扩展器的拓扑唯一性

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-02-07 DOI: 10.4310/cjm.2022.v10.n4.a2

J. Bernstein, Lu Wang

引用次数: 20

Mixed-$textrm{Spin-P}$ fields of Fermat polynomials 费马多项式的混合-$textrm{Spin-P}$域

IF 1.6 2区数学

Cambridge Journal of Mathematics Pub Date : 2019-01-01 DOI: 10.4310/cjm.2019.v7.n3.a3

Huai-liang Chang, Jun Li, Wei-Ping Li, Melissa Chiu-Chu Liu

引用次数: 1